was made from sheet steel. What is the minimum number of square meters of sheet metal used to make it? Think about the answer!","","Um die Oberfläche eines kugelförmigen Behälters zu berechnen, können wir die Formel für die Oberfläche einer Kugel verwenden:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 4\\pi r^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Gegeben:\u003Cbr />Durchmesser, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>d = 49\u003C/math-field>\u003C/math-field> m\u003Cbr />\u003Cbr />Wir können den Radius \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$r$\u003C/math-field>\u003C/math-field> berechnen, da \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>r = \\frac{d}{2}\u003C/math-field>\u003C/math-field>:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = \\frac{49}{2} = 24.5 \\text{ m} \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Nun können wir die Formel für die Oberfläche verwenden:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 4\\pi \\times $24.5$^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 4\\pi \\times 24.5^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 4\\pi \\times 600.25 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A \\approx 2401.00 \\text{ m}^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Antwort: Bei der Herstellung des kugelförmigen Gasbehälters wurden mindestens \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2401.00 \\, \\text{m}^2\u003C/math-field>\u003C/math-field> Stahlblech verarbeitet.",1043,209,"task-9-a-spherical-gas-tank-d-49-m-was-made-from-sheet-steel-what-is-the-minimum-number-of-square-meters-of-sheet-metal-used-to-make-it-think-about-the-answer",{"id":44,"category":36,"text_question":45,"photo_question":38,"text_answer":46,"step_text_answer":8,"step_photo_answer":8,"views":47,"likes":48,"slug":49},536001,"$2ax³$ $8x³y-3y²$","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=2ax^{3}\\cdot 8x^{3}y-2ax^{3}\\cdot 3y^{2}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=16ax^{6}y-6ax^{3}y^{2}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1480,296,"2ax-8x-y-3y",{"id":51,"category":36,"text_question":52,"photo_question":38,"text_answer":53,"step_text_answer":8,"step_photo_answer":8,"views":54,"likes":55,"slug":56},536000,"Bonga currently has a portfolio of ordinary shares representing several different companies. Bonga considers it to be a well-balanced investment portfolio, but he wants to reduce the overall risk of the portfolio a bit more by including ordinary shares from Titan Mining Corporation.\r\nThe following information on Titan Mining Corporation is available:\r\nFor the period 2017 to 2020, the company paid the following dividends per year respectively: R3,14; R3,55; R3,89; and R3,95. The 2021 dividend is expected to increase by the average growth rate of the dividends between 2017 and 2020, and the dividend will increase by 10% per year indefinitely from 2022 onwards.\r\nBonga requires a return of 15% on his investment portfolio and is not prepared to pay more than R52,00 per ordinary share of Titan Mining Corporation.Calculate the current price of Titan Mining Corporation’s ordinary share.","\"To calculate the current price of Titan Mining Corporation's ordinary share, we can use the Dividend Discount Model $DDM$, which is particularly useful for companies that pay dividends regularly. The formula for the price of a stock in this model is:\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> P = \\frac{D_1}{r - g} \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />where:\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> P \u003C/math-field>\u003C/math-field> is the current price of the stock.\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> D_1 \u003C/math-field>\u003C/math-field> is the expected dividend next year.\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r \u003C/math-field>\u003C/math-field> is the required rate of return.\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> g \u003C/math-field>\u003C/math-field> is the expected growth rate of dividends.\u003Cbr />\u003Cbr />### Step 1: Calculate the Average Growth Rate from 2017 to 2020\u003Cbr />\u003Cbr />The growth rates for each year are as follows:\u003Cbr />- From 2017 to 2018: 13.06%\u003Cbr />- From 2018 to 2019: 9.58%\u003Cbr />- From 2019 to 2020: 1.54%\u003Cbr />\u003Cbr />The average annual growth rate over the period from 2017 to 2020 is approximately 8.06%.\u003Cbr />\u003Cbr />### Step 2: Calculate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> D_1 \u003C/math-field>\u003C/math-field> for 2021\u003Cbr />\u003Cbr />Given the average growth rate obtained in Step 1, the dividend for 2021 $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> D_1 \u003C/math-field>\u003C/math-field>$ is calculated using:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> D_1 = \\text{Dividend}_{2020} \\times $1 + \\text{average growth rate}$ \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />### Step 3: Calculate the Price of the Stock Using DDM\u003Cbr />\u003Cbr />Using a required return \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = 0.15 \u003C/math-field>\u003C/math-field> $15%$ and a growth rate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> g = 0.10 \u003C/math-field>\u003C/math-field> $10%$ from 2022 onwards, the price of the stock is calculated as:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> P = \\frac{D_1}{0.15 - 0.10} \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />### Calculation Details\u003Cbr />\u003Cbr />- Dividend for 2021 $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> D_1 \u003C/math-field>\u003C/math-field>$: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> R4.27 \u003C/math-field>\u003C/math-field>\u003Cbr />- Price of the Stock Using DDM: Approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> R85.37 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{Current \\ price = R85.37}\u003C/math-field>\u003C/math-field>\"",1188,238,"bonga-currently-has-a-portfolio-of-ordinary-shares-representing-several-different-companies-bonga-considers-it-to-be-a-well-balanced-investment-portfolio-but-he-wants-to-reduce-the-overall-risk-of-t",{"id":58,"category":36,"text_question":59,"photo_question":38,"text_answer":60,"step_text_answer":8,"step_photo_answer":8,"views":61,"likes":62,"slug":63},535999,"5:x=8:9\n Proportions","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$\\begin{array} { l }\\frac{ 5 }{ x }=\\frac{ 8 }{ 9 },& x≠0\\end{array}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$45=8x$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$8x=45$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$\\begin{array} { l }x=\\frac{ 45 }{ 8 },& x≠0\\end{array}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$\\begin{align*}&x=\\frac{ 45 }{ 8 } \\\\&\\begin{array} { l }x=5 \\frac{ 5 }{ 8 },& x=5.625\\end{array}\\end{align*}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1314,263,"5-x-8-9-proportions",{"id":65,"category":36,"text_question":66,"photo_question":38,"text_answer":67,"step_text_answer":8,"step_photo_answer":8,"views":68,"likes":69,"slug":70},535998,"b) 3 + 1/7 =","\u003Cmath-field read-only> 3+\\frac{1}{7}=\\frac{21+1}{7}=\\frac{22}{7} \u003C/math-field>",908,182,"b-3-1-7",{"id":72,"category":36,"text_question":73,"photo_question":38,"text_answer":74,"step_text_answer":8,"step_photo_answer":8,"views":75,"likes":76,"slug":77},535997,"[$-3/2$ – $6/3$] ÷ $-8/2$","Calculate the difference \n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$\\frac{ -\\frac{ 7 }{ 2 } }{ -\\frac{ 8 }{ 2 } }$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n Cancel out the common factor \u003Cmath-field read-only default-mode=\"inline-math\" style=\"padding-left:5px; padding-right:5px; display:inline-block; border-radius:4px; border: 1px solid rgba$0, 0, 0, .3$\">$2$\u003C/math-field> \n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$\\frac{ -7 }{ -8 }$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n A fraction with a negative numerator and negative denominator is always positive \n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$\\begin{align*}&\\frac{ 7 }{ 8 } \\\\&0.875\\end{align*}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1276,255,"3-2-6-3-8-2",{"id":79,"category":36,"text_question":80,"photo_question":38,"text_answer":81,"step_text_answer":8,"step_photo_answer":8,"views":82,"likes":83,"slug":84},535995,"If a group of distinct objects can be arranged in 120 different ways, how many objects are there?","ANSWER:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">$5$\u003C/math-field>\u003Cbr>\u003Cbr>STEP-BY-STEP EXPLANATION:\u003Cbr>\u003Cbr>n! = 120\u003Cbr>\u003Cbr>n = 5",1272,254,"if-a-group-of-distinct-objects-can-be-arranged-in-120-different-ways-how-many-objects-are-there",{"id":86,"category":36,"text_question":87,"photo_question":38,"text_answer":88,"step_text_answer":8,"step_photo_answer":8,"views":89,"likes":90,"slug":91},535994,"Determine the elements of the parabola and graph, whose general equation is: x2 - 12x + 10y - 4= 0","Para determinar los elementos de la parábola con la ecuación general \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^2 - 12x + 10y - 4 = 0\u003C/math-field>\u003C/math-field>, primero necesitamos convertirla a su forma estándar completa. La ecuación general de una parábola es de la forma \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Ax^2 + Bx + Cy + D = 0\u003C/math-field>\u003C/math-field>.\u003Cbr />\u003Cbr />Dada la ecuación \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^2 - 12x + 10y - 4 = 0\u003C/math-field>\u003C/math-field>, vamos a completar el cuadrado para \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> y despejar \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field>:\u003Cbr />\u003Cbr />1. Pasamos los términos con \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> a un lado y los términos independientes al otro lado de la ecuación:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^2 - 12x + 10y = 4\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />2. Completamos el cuadrado para \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$x^2 - 12x + 36$ + 10y = 4 + 36\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$x - 6$^2 + 10y = 40\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />3. Despejamos \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field>:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10y = - $x - 6$^2 + 40\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = -\\frac{1}{10}$x - 6$^2 + 4\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Ahora que hemos convertido la ecuación general a la forma estándar completa, podemos identificar los elementos de la parábola:\u003Cbr />\u003Cbr />- Vértice: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$h, k$ = $6, 4$\u003C/math-field>\u003C/math-field>\u003Cbr />- Eje de simetría: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 6\u003C/math-field>\u003C/math-field>\u003Cbr />- Distancia focal: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>|p| = \\frac{1}{4a} = \\frac{1}{4$1/10$} = 2.5\u003C/math-field>\u003C/math-field>\u003Cbr />- Foco: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$h, k + p$ = $6, 4 + 2.5$ = $6, 6.5$\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Graficando la parábola:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\begin{array}{ c }\\begin{tikzpicture} \\begin{axis}[ axis lines = center, xlabel = \u003C/math-field>\u003C/math-field>x\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>, ylabel = \u003C/math-field>\u003C/math-field>y\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>, xmin = 0, xmax = 12, ymin = 0, ymax = 10, ] \\addplot [ domain=0:12, samples=100, color=blue, ] {-1/10*((x-6)^2)+4}; \\addplot [only marks, mark=*] coordinates {(6,4) (6,6.5)}; \\end{axis}\\end{tikzpicture}\\end{array}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\textbf{Respuesta: }\u003C/math-field>\u003C/math-field> La parábola tiene vértice en el punto \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(6, 4)\u003C/math-field>\u003C/math-field>, eje de simetría en \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 6\u003C/math-field>\u003C/math-field>, una distancia focal de 2.5 y su foco se encuentra en el punto \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(6, 6.5)\u003C/math-field>\u003C/math-field>.",710,142,"determine-the-elements-of-the-parabola-and-graph-whose-general-equation-is-x2-12x-10y-4-0",{"id":93,"category":36,"text_question":94,"photo_question":38,"text_answer":95,"step_text_answer":8,"step_photo_answer":8,"views":96,"likes":97,"slug":98},535992,"Show that the derivative of the complex logarithm of a z belonging to C is edale at 1/z","Soit \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z \\in \\mathbb{C}\u003C/math-field>\u003C/math-field> . On sait que la fonction logarithme complexe est définie par: \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log(z) = \\ln|z| + i \\arg(z)\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>où \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\ln|z|\u003C/math-field>\u003C/math-field> est le logarithme népérien du module de \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> et \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\arg(z)\u003C/math-field>\u003C/math-field> est l'argument de \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> . \u003Cbr>\u003Cbr>Calculons maintenant la dérivée de \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log(z)\u003C/math-field>\u003C/math-field> par rapport à \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> :\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{d}{dz}(\\log(z)) = \\frac{d}{dz}(\\ln|z|) + i \\frac{d}{dz}(\\arg(z))\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Puisque \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\ln|z|\u003C/math-field>\u003C/math-field> ne dépend que du module de \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> et que \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\arg(z)\u003C/math-field>\u003C/math-field> ne dépend que de l'argument de \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> , les dérivées par rapport à \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> de ces deux parties sont nulles. Ainsi, on obtient :\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{d}{dz}(\\log(z)) = 0 + i \\cdot 0 = 0\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Donc la dérivée du logarithme complexe de \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> par rapport à \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z\u003C/math-field>\u003C/math-field> est nulle. \u003Cbr>\u003Cbr>\\textbf{Réponse:} \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{d}{dz}(\\log(z)) = 0\u003C/math-field>\u003C/math-field>",1281,256,"show-that-the-derivative-of-the-complex-logarithm-of-a-z-belonging-to-c-is-edale-at-1-z",{"id":100,"category":36,"text_question":101,"photo_question":38,"text_answer":102,"step_text_answer":8,"step_photo_answer":8,"views":103,"likes":104,"slug":105},535991,"Write the equation of the line with a\nSlope of 2 and passes through the point (-1, 5).","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$y=2x+7$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1473,295,"write-the-equation-of-the-line-with-a-slope-of-2-and-passes-through-the-point-1-5",{"id":107,"category":36,"text_question":108,"photo_question":38,"text_answer":109,"step_text_answer":8,"step_photo_answer":8,"views":110,"likes":111,"slug":112},535990,"You have $50,000 in savings for retirement in an investment earning 9% annually. You aspire to have $1,000,000 in savings when you retire. Assuming you add no more to your savings, how many years will it take to reach your goal?\n\nPlease round your answer to the nearest hundredth.\n\nNote that the HP 12c financial calculator rounds up the periods result to the next integer and will not give the correct answer to the nearest hundredth. Therefore, you should use Excel or a financial calculator that does provide decimal precision to the number of periods.","To calculate the number of years needed to reach the savings goal, we can use the future value formula for compound interest:\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>FV = PV \\times (1 + r)^n\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>where:\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>FV\u003C/math-field>\u003C/math-field> is the future value you want to achieve (\\$1,000,000),\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>PV\u003C/math-field>\u003C/math-field> is the present value (initial savings, $50,000),\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>r\u003C/math-field>\u003C/math-field> is the annual interest rate (9% or 0.09), and\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n\u003C/math-field>\u003C/math-field> is the number of years.\u003Cbr>\u003Cbr>Plugging in the given values, we have:\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\50,000 \\times (1 + 0.09)^n\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>20 = (1.09)^n\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>To solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n\u003C/math-field>\u003C/math-field> , we will use logarithms:\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n = \\frac{{\\ln(20)}}{{\\ln(1.09)}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n\\approx34.76\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, it will take approximately 34.76 years to reach $1,000,000 in savings, rounded to the nearest hundredth.\u003Cbr>",375,75,"you-have-50-000-in-savings-for-retirement-in-an-investment-earning-9-annually-you-aspire-to-have-1-000-000-in-savings-when-you-retire-assuming-you-add-no-more-to-your-savings-how-many-years-will",{"id":114,"category":36,"text_question":115,"photo_question":38,"text_answer":116,"step_text_answer":8,"step_photo_answer":8,"views":117,"likes":90,"slug":118},535989,"Establish a linear equation that expresses the relationship between temperature in\n degrees Celsius, C, and degrees Kelvin, K. Use the fact that water freezes at 0o\n Celsius (273.15 Kelvin) and boils at 100o Celsius (373.15 Kelvin).","Let's use the two data points provided to establish a linear equation relating Celsius temperature, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>C\u003C/math-field>\u003C/math-field>, to Kelvin temperature, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>K\u003C/math-field>\u003C/math-field>.\u003Cbr />\u003Cbr />Given data points:\u003Cbr />1. Water freezes at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0^\\circ C\u003C/math-field>\u003C/math-field> which is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>273.15\u003C/math-field>\u003C/math-field> Kelvin.\u003Cbr />2. Water boils at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>100^\\circ C\u003C/math-field>\u003C/math-field> which is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>373.15\u003C/math-field>\u003C/math-field> Kelvin.\u003Cbr />\u003Cbr />Let's find the slope, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>m\u003C/math-field>\u003C/math-field>, of the linear equation using the two points:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>m = \\frac{K_2 - K_1}{C_2 - C_1} = \\frac{373.15 - 273.15}{100 - 0} = \\frac{100}{100} = 1\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Now that we have the slope, we can write the point-slope form of a linear equation:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>K - K_1 = m(C - C_1)\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>K - 273.15 = 1(C - 0)\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>K - 273.15 = C\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Therefore, the linear equation relating Celsius temperature to Kelvin temperature is:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>K = C + 273.15\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{K = C + 273.15}\u003C/math-field>\u003C/math-field>",708,"establish-a-linear-equation-that-expresses-the-relationship-between-temperature-in-degrees-celsius-c-and-degrees-kelvin-k-use-the-fact-that-water-freezes-at-0o-celsius-273-15-kelvin-and-boils",{"id":120,"category":36,"text_question":121,"photo_question":38,"text_answer":122,"step_text_answer":8,"step_photo_answer":8,"views":123,"likes":124,"slug":125},535988,"Annual salaries for graduates are expected to be between $30,000 and $45,000. Assume that a 95% confidence interval estimate of the population mean annual sling salary is desired. How large a sample should be taken if the desired margin of error is $250.","**Step 1:** \u003Cbr />Given parameters: \u003Cbr />Desired Margin of Error (ME): $250 \u003Cbr />Confidence Level: 95% \u003Cbr />\u003Cbr />**Step 2:** \u003Cbr />Calculate the estimated population standard deviation (\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sigma \u003C/math-field>\u003C/math-field>): \u003Cbr />Given range: $30,000 to $45,000 which covers approximately 6 standard deviations \u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sigma = \\frac{Range}{6} = \\frac{45000 - 30000}{6} = \\frac{15000}{6} = 2500\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />**Step 3:** \u003Cbr />Calculate the required sample size (\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> n \u003C/math-field>\u003C/math-field>): \u003Cbr />Substitute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\sigma \u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> z \u003C/math-field>\u003C/math-field> into the formula for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> n \u003C/math-field>\u003C/math-field>: \u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n = \\left(\\frac{1.96 \\times 2500}{250}\\right)^2\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Calculating step by step: \u003Cbr />1. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1.96 \\times 2500}{250} = 19.6 \u003C/math-field>\u003C/math-field> \u003Cbr />2. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 19.6^2 = 384.16 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Therefore, approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\boxed{384} \u003C/math-field>\u003C/math-field> participants would be required for the sample.",458,92,"annual-salaries-for-graduates-are-expected-to-be-between-30-000-and-45-000-assume-that-a-95-confidence-interval-estimate-of-the-population-mean-annual-sling-salary-is-desired-how-large-a-sample-s",{"id":127,"category":36,"text_question":128,"photo_question":38,"text_answer":129,"step_text_answer":8,"step_photo_answer":8,"views":130,"likes":131,"slug":132},535987,"A 0.5 kg piece of ice at -10 degrees Celsius is placed in 3 kg of water at 20 degrees Celsius.\n\n What temperature will the mixture finally be at?\n\n Consider that: cA=4186 J/kg°C and cH=2100 J/kg°C","Step 1: Calculate the heat required to raise the temperature of ice from -10°C to 0°C.\u003Cbr>\u003Cbr>The heat (Q) required to raise the temperature of the ice can be calculated using the formula:\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = mcΔT\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Given: mass of ice (m) = 0.5 kg, specific heat of ice (cA) = 2100 J/kg°C, initial temperature (Ti) = -10°C, final temperature (Tf) = 0°C\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = 0.5 \\times 2100 \\times (0 - (-10))\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = 0.5 \\times 2100 \\times 10 = 10500 J\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 2: Calculate the heat required to melt the ice at 0°C.\u003Cbr>\u003Cbr>The heat (Q) required to melt the ice can be calculated using the formula:\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = mL\u003C/math-field>\u003C/math-field> \u003Cbr> step 2.1:calculate the heat required to raise water(from ice) to final temperature\u003Cbr>Q=0.5*4186*x\u003Cbr>\u003Cbr>\u003Cbr>Given: latent heat of fusion of ice (L) = 334000 J/kg\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = 0.5 \\times 334000 = 167000 J\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 3: Calculate the heat required to raise the temperature of the water from 20°C to the final temperature.\u003Cbr>\u003Cbr>The heat (Q) required to raise the temperature of water can be calculated using the formula:\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = mcΔT\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Given: mass of water (m) = 3 kg, specific heat of water (cH) = 4186 J/kg°C, initial temperature (Ti) = 20°C, final temperature (Tf) = x°C\u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = 3 \\times 4186 \\times (x - 20)\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 4: The total heat added to the system must be equal to zero for the final temperature to be reached. \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10500+167000+0.5\\times4186\\times x+3\\times4186\\times(x-20)=0\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Solving the equation gives: \u003Cbr> \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>177500+14651x-251160=0\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr> \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14651x=73660\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x ≈ 5°C\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the final temperature of the mixture will be approximately 5°C.\u003Cbr>\u003Cbr>\\textbf{Answer:} The final temperature of the mixture will be approximately 5°C.",1354,271,"a-0-5-kg-piece-of-ice-at-10-degrees-celsius-is-placed-in-3-kg-of-water-at-20-degrees-celsius-what-temperature-will-the-mixture-finally-be-at-consider-that-ca-4186-j-kg-c-and-ch-2100-j-kg-c",{"id":134,"category":36,"text_question":135,"photo_question":38,"text_answer":136,"step_text_answer":8,"step_photo_answer":8,"views":137,"likes":138,"slug":139},535986,"Calculate the value of the function f(x)=3x+4x considering that \"x\" is equal to -4","f(x) = 3x + 4x = 7x\n f(x) = 7x\n cuando x = -4\n f(-4) = 7 * -4 = -28",425,85,"calculate-the-value-of-the-function-f-x-3x-4x-considering-that-x-is-equal-to-4",{"id":141,"category":36,"text_question":142,"photo_question":38,"text_answer":143,"step_text_answer":8,"step_photo_answer":8,"views":144,"likes":145,"slug":146},535985,"Calculate the value of \"x\" in the following equation x+2x+3x-5=4x-9","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$6x-5=4x-9$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$6x-5-4x+9=0$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$2x-5+9=0$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$2x+4=0$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$x+2=0;x=-2$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1426,285,"calculate-the-value-of-x-in-the-following-equation-x-2x-3x-5-4x-9",{"id":148,"category":36,"text_question":149,"photo_question":38,"text_answer":150,"step_text_answer":8,"step_photo_answer":8,"views":151,"likes":152,"slug":153},535984,"In an English course there are 16 men and 24 women, only four men wear long hair, the rest wear it short, while only six women wear short hair, the rest wear it long, what is the probability that the person has long hair? long?","To find the probability that a randomly chosen person has long hair, we need to find the total number of people with long hair and divide it by the total number of people in the class.\u003Cbr>\u003Cbr>Let's find the total number of people with long hair:\u003Cbr>- Number of men with long hair = 4\u003Cbr>- Number of women with long hair = 24 - 6 = 18\u003Cbr>\u003Cbr>Total number of people with long hair = 4 (men) + 18 (women) = 22\u003Cbr>\u003Cbr>Next, let's find the total number of people in the class:\u003Cbr>Total number of men = 16\u003Cbr>Total number of women = 24\u003Cbr>\u003Cbr>Total number of people in the class = 16 (men) + 24 (women) = 40\u003Cbr>\u003Cbr>Now, let's find the probability that a person has long hair:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(\\text{long hair}) = \\dfrac{\\text{Total number of people with long hair}}{\\text{Total number of people in the class}}\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(\\text{long hair})=\\dfrac{22}{40}=\\frac{11}{20}\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(\\text{long hair}) = 0.55\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the probability that a randomly chosen person has long hair is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{0.55}\u003C/math-field>\u003C/math-field> .",1033,207,"in-an-english-course-there-are-16-men-and-24-women-only-four-men-wear-long-hair-the-rest-wear-it-short-while-only-six-women-wear-short-hair-the-rest-wear-it-long-what-is-the-probability-that-the",{"id":155,"category":36,"text_question":156,"photo_question":38,"text_answer":157,"step_text_answer":8,"step_photo_answer":8,"views":158,"likes":159,"slug":160},535983,"In the lioness vicar house group I contribute the amount proportional to the number of students they have. If each group contributed 956 for a school event.\n What percentage did the students contribute in each group, considering that.\n 1A has 31 Students\n 1B has 34 Students\n 2A has 20 Students\n 2B has 22 Students\n 3A has 18 Students\n 3B has 24 Students\n How much does each student contribute in their group?\n What percentage does it correspond to?\n How much money did the entire school raise in total?","Let's first find out how much each student contributed in their group.\u003Cbr>\u003Cbr>Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> be the amount each student contributes in their group.\u003Cbr>\u003Cbr>For group 1A with 31 students, the total contribution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>31x = 956\u003C/math-field>\u003C/math-field> . So, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{956}{31}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>For group 1B with 34 students, the total contribution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>34x = 956\u003C/math-field>\u003C/math-field> . So, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{956}{34}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>For group 2A with 20 students, the total contribution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>20x = 956\u003C/math-field>\u003C/math-field> . So, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{956}{20}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>For group 2B with 22 students, the total contribution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>22x = 956\u003C/math-field>\u003C/math-field> . So, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{956}{22}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>For group 3A with 18 students, the total contribution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>18x = 956\u003C/math-field>\u003C/math-field> . So, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{956}{18}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>For group 3B with 24 students, the total contribution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>24x = 956\u003C/math-field>\u003C/math-field> . So, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{956}{24}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>Now, let's find out the percentage each student contributed in their group. Since each student contributed the same amount in their group, we will use the values of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> calculated for each group.\u003Cbr>\u003Cbr>The percentage each student contributed in their group is given by: \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left( \\frac{x}{956} \\right) \\times 100 \\%\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Now, let's calculate the total amount raised by the entire school.\u003Cbr>Total amount raised \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 1A + 1B + 2A + 2B + 3A + 3B = 6 \\times 956\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>### Answer\u003Cbr>\u003Cbr>1. Each student contributes:\u003Cbr> - Group 1A: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{956}{31}=30.84\u003C/math-field>\u003C/math-field> dollars\u003Cbr> - Group 1B: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{956}{34}=28.12\u003C/math-field>\u003C/math-field> dollars\u003Cbr> - Group 2A: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{956}{20}=47.8\u003C/math-field>\u003C/math-field> dollars\u003Cbr> - Group 2B: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{956}{22}=43.45\u003C/math-field>\u003C/math-field> dollars\u003Cbr> - Group 3A: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{956}{18}=53.11\u003C/math-field>\u003C/math-field> dollars\u003Cbr> - Group 3B: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{956}{24}=39.83\u003C/math-field>\u003C/math-field> dollars\u003Cbr>\u003Cbr>2. The percentage each student contributed in their group:\u003Cbr> - Group 1A: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left(\\frac{956}{31 \\times956}\\right)\\times100\\%=3.23\\%\u003C/math-field>\u003C/math-field> \u003Cbr> - Group 1B: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left(\\frac{956}{34 \\times956}\\right)\\times100\\%=2.94\\%\u003C/math-field>\u003C/math-field> \u003Cbr> - Group 2A: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left(\\frac{956}{20 \\times956}\\right)\\times100\\%=5\\%\u003C/math-field>\u003C/math-field> \u003Cbr> - Group 2B: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left(\\frac{956}{22 \\times956}\\right)\\times100\\%=4.55\\%\u003C/math-field>\u003C/math-field> \u003Cbr> - Group 3A: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left(\\frac{956}{18 \\times956}\\right)\\times100\\%=5.56\\%\u003C/math-field>\u003C/math-field> \u003Cbr> - Group 3B: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left(\\frac{956}{24 \\times956}\\right)\\times100\\%=4.17\\%\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Total amount raised by the entire school: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>6 \\times 956 = 5736\u003C/math-field>\u003C/math-field> dollars",1377,275,"in-the-lioness-vicar-house-group-i-contribute-the-amount-proportional-to-the-number-of-students-they-have-if-each-group-contributed-956-for-a-school-event-what-percentage-did-the-students-contribut",{"id":162,"category":36,"text_question":163,"photo_question":38,"text_answer":164,"step_text_answer":8,"step_photo_answer":8,"views":165,"likes":166,"slug":167},535982,"1. Administer 10 mg of Morphine every 4 hours in 100 ml of physiological solution.\n Presentations: 2.5mg vial in 2.5ml. 50mg vial in 2ml and 10mg vial in 10ml\n How many ml of physiological solution will there be and how many ml of Morphine using the presentation of 2.5mg in 2.5ml?\n How many vials should you use with the presentation of 2.5mg in 2.5ml?\n How many ml of physiological solution will there be and how many ml of Morphine using the presentation of 50mg in 2ml?\n How many ml of physiological solution will there be and how many ml of Morphine using the presentation of 10mg in 10ml?","Let's start with the calculations for each presentation:\u003Cbr />\u003Cbr />### Presentation 1: 2.5mg in 2.5ml\u003Cbr />\u003Cbr />1. Dose requirement: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{mg}\u003C/math-field>\u003C/math-field>\u003Cbr />2. Volume requirement per dose using 2.5 mg vial: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{mg} \\div 2.5 \\, \\text{mg/vial} = 4 \\, \\text{vials}\u003C/math-field>\u003C/math-field>\u003Cbr />3. Morphine volume for the dose: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4 \\times 2.5 \\, \\text{ml} = 10 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />4. Volume of physiological solution: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>100 \\, \\text{ml} - 10 \\, \\text{ml} = 90 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />**Answers for Presentation 1:**\u003Cbr />- Morphine volume: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />- Physiological solution volume: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>90 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />### Presentation 2: 50mg in 2ml\u003Cbr />\u003Cbr />1. Dose requirement: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{mg}\u003C/math-field>\u003C/math-field>\u003Cbr />2. Calculation of Morphine volume needed: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{10 \\, \\text{mg}}{50 \\, \\text{mg}} \\times 2 \\, \\text{ml} = 0.4 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />3. Volume of physiological solution: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>100 \\, \\text{ml} - 0.4 \\, \\text{ml} = 99.6 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />**Answers for Presentation 2:**\u003Cbr />- Morphine volume: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.4 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />- Physiological solution volume: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>99.6 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />### Presentation 3: 10mg in 10ml\u003Cbr />\u003Cbr />1. Dose requirement: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{mg}\u003C/math-field>\u003C/math-field>\u003Cbr />2. Morphine volume for the dose: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />3. Volume of physiological solution: No additional physiological solution needed\u003Cbr />\u003Cbr />**Answers for Presentation 3:**\u003Cbr />- Morphine volume: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />- Physiological solution volume: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>90 \\, \\text{ml}\u003C/math-field>\u003C/math-field> (to keep the total administration volume at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>100 \\, \\text{ml}\u003C/math-field>\u003C/math-field>)\u003Cbr />\u003Cbr />Therefore, the calculations using the three different presentations of Morphine have shown the Morphine and physiological solution volumes required for each scenario. \u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\textbf{Final answers:}\u003C/math-field>\u003C/math-field>\u003Cbr />- For Presentation 1: Morphine volume - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{ml}\u003C/math-field>\u003C/math-field>, Physiological solution volume - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>90 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />- For Presentation 2: Morphine volume - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.4 \\, \\text{ml}\u003C/math-field>\u003C/math-field>, Physiological solution volume - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>99.6 \\, \\text{ml}\u003C/math-field>\u003C/math-field>\u003Cbr />- For Presentation 3: Morphine volume - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10 \\, \\text{ml}\u003C/math-field>\u003C/math-field>, Physiological solution volume - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>90 \\, \\text{ml}\u003C/math-field>\u003C/math-field>",635,127,"1-administer-10-mg-of-morphine-every-4-hours-in-100-ml-of-physiological-solution-presentations-2-5mg-vial-in-2-5ml-50mg-vial-in-2ml-and-10mg-vial-in-10ml-how-many-ml-of-physiological-solution-wi",{"id":169,"category":36,"text_question":170,"photo_question":38,"text_answer":171,"step_text_answer":8,"step_photo_answer":8,"views":172,"likes":173,"slug":174},535981,"What is the general solution of the differential y'= y/(x+1)","Given differential equation is: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{dy}{dx} = \\frac{y}{x+1}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>Step 1: Separate the variables:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{dy}{y} = \\frac{dx}{x+1}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 2: Integrate both sides:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int\\frac{1}{y} dy = \\int\\frac{1}{x+1} dx\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 3: Solve the integrals:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>ln|y| = ln|x+1| + C\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 4: Eliminate the natural logarithms: \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>e^{ln|y|} = e^{ln|x+1| + C}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 5: Simplify the equation:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = e^C \\cdot |x + 1|\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 6: Finally, rewrite the arbitrary constant as \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>e^C = C_1\u003C/math-field>\u003C/math-field> :\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{y = C_1 \\cdot |x + 1|}\u003C/math-field>\u003C/math-field> , where \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>C_1\u003C/math-field>\u003C/math-field> is an arbitrary constant.",630,126,"what-is-the-general-solution-of-the-differential-y-y-x-1",{"first":6,"last":176,"prev":177,"next":124},188,90,{"current_page":179,"from":180,"last_page":176,"links":181,"path":215,"per_page":216,"to":217,"total":218},91,1801,[182,185,187,189,192,195,197,200,202,205,208,211,213],{"url":177,"label":183,"active":184},"« Previous",false,{"url":6,"label":186,"active":184},"1",{"url":10,"label":188,"active":184},"2",{"url":190,"label":191,"active":184},88,"88",{"url":193,"label":194,"active":184},89,"89",{"url":177,"label":196,"active":184},"90",{"url":179,"label":198,"active":199},"91",true,{"url":124,"label":201,"active":184},"92",{"url":203,"label":204,"active":184},93,"93",{"url":206,"label":207,"active":184},94,"94",{"url":209,"label":210,"active":184},187,"187",{"url":176,"label":212,"active":184},"188",{"url":124,"label":214,"active":184},"Next »","https://api.math-master.org/api/question",20,1820,3742,{"data":220},{"questions":221},[222,226,230,234,238,242,246,250,254,258,262,266,270,274,278,282,286,290,294,298],{"id":223,"category":36,"text_question":224,"slug":225},532026,"Simplify the expression sin³(x)+cos³(x), using trigonometric functions","simplify-the-expression-sin-x-cos-x-using-trigonometric-functions",{"id":227,"category":36,"text_question":228,"slug":229},533909,"Express the following numbers in decimal system,\n where the subscript indicates the base: 110101 (SUBINDEX=2)","express-the-following-numbers-in-decimal-system-where-the-subscript-indicates-the-base-110101-subindex-2",{"id":231,"category":36,"text_question":232,"slug":233},533953,"Using the integration by parts method, calculate the integral of [x².ln(1/x)]dx:\n x\n 4\n /4\n x³/6\n x\n 4\n /8\n x³/3\n x\n 4\n /6","using-the-integration-by-parts-method-calculate-the-integral-of-x-ln-1-x-dx-x-4-4-x-6-x-4-8-x-3-x-4-6",{"id":235,"category":36,"text_question":236,"slug":237},534007,"A soft drink machine outputs a mean of 23 ounces per cup. The machines output is normally distributed with a standard deviation of 3 ounces. What is the probability of filling a cup between 26 and 28 ounces round your answer to four decimal places","a-soft-drink-machine-outputs-a-mean-of-23-ounces-per-cup-the-machines-output-is-normally-distributed-with-a-standard-deviation-of-3-ounces-what-is-the-probability-of-filling-a-cup-between-26-and-28",{"id":239,"category":36,"text_question":240,"slug":241},534021,"The beta of a company is 1.51 while its financial leverage is 27%. What is then its unlevered beta if the corporate tax rate is 40%? (4 decimal places)","the-beta-of-a-company-is-1-51-while-its-financial-leverage-is-27-what-is-then-its-unlevered-beta-if-the-corporate-tax-rate-is-40-4-decimal-places",{"id":243,"category":36,"text_question":244,"slug":245},534073,"How long will it take for $900 to become $5000 at an annual rate of 11.15% compounded bimonthly?","how-long-will-it-take-for-900-to-become-5000-at-an-annual-rate-of-11-15-compounded-bimonthly",{"id":247,"category":36,"text_question":248,"slug":249},534140,"The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.\n\n84. Find the probability that the average price for 30 gas stations is less than $4.55.\n\ta\t0.6554\n\tb\t0.3446\n\tc\t0.0142\n\td\t0.9858\n\te\t0","the-cost-of-unleaded-gasoline-in-the-bay-area-once-followed-an-unknown-distribution-with-a-mean-of-4-59-and-a-standard-deviation-of-0-10-sixteen-gas-stations-from-the-bay-area-are-randomly-chosen",{"id":251,"category":36,"text_question":252,"slug":253},534165,"You mix a powder drug with a 4.5ml of liquid to get a reconstituted solution with a concentration of 250mg/ml. The prescribers order is for 500 mg . You will give what ml of the reconstituted solution","you-mix-a-powder-drug-with-a-4-5ml-of-liquid-to-get-a-reconstituted-solution-with-a-concentration-of-250mg-ml-the-prescribers-order-is-for-500-mg-you-will-give-what-ml-of-the-reconstituted-solutio",{"id":255,"category":36,"text_question":256,"slug":257},534232,"Your boss asks you to plan the sample size for a randomized, double-blind, controlled trial in\nthe clinical development of a cure for irritable bowl disease. Current standard treatment shall be\ncompared with a new treatment in this trial. The S3-guideline of AWM demonstrated a mean\nchange of the summary score of the validated health related quality of life questionnaire at 8\nweeks of 16 with standard deviation 23 under standard treatment. You quote the drop-out rate\nof 11% from literature (previous phase of clinical development). Your research yielded a clinically\nimportant effect of 4 that has been found to be the Minimal Clinically Important Difference (MCID).\nIn order to demonstrate superiority of the new treatment over standard of care, you assume that\nthe change in of the summary score of the validated health related quality of life questionnaire\nfollows a normal distribution, and that the standard deviation is the same for both treatments.\nHow many patientes would one need to recruit for the trial to demonstrate the clinically\ninteresting difference between treatments at significance level 5% with 95% power?","your-boss-asks-you-to-plan-the-sample-size-for-a-randomized-double-blind-controlled-trial-in-the-clinical-development-of-a-cure-for-irritable-bowl-disease-current-standard-treatment-shall-be-compar",{"id":259,"category":36,"text_question":260,"slug":261},534345,"On+January+10+2023+the+CONSTRUCTORA+DEL+ORIENTE+SAC+company+acquires+land+to+develop+a+real estate+project%2C+which+prev%C3% A9+enable+50+lots+for+commercial+use+valued+in+S%2F+50%2C000.00+each+one%2C+the+company+has+as+a+business+model+generate+ cash+flow+through%C3%A9s+of+the+rental%2C+so+47%2C+of+the+50+enabled+lots+are+planned to lease+47%2C+and+ the+rest+will be%C3%A1n+used+by+the+company+for+management%C3%B3n+and+land+control","on-january-10-2023-the-constructora-del-oriente-sac-company-acquires-land-to-develop-a-real-estate-project-2c-which-prev-c3-a9-enable-50-lots-for-commercial-use-valued-in-s-2f-50-2c000-00-each-one-2c",{"id":263,"category":36,"text_question":264,"slug":265},534363,"ind the z-score for which 72% of the distribution's area lies between -z and z.\n\n-1.7417, 1.7417\n\n-1.1538, 1.1538\n\n-1.0803, 1.0803\n\n-2.826, 2.826","ind-the-z-score-for-which-72-of-the-distribution-s-area-lies-between-z-and-z-1-7417-1-7417-1-1538-1-1538-1-0803-1-0803-2-826-2-826",{"id":267,"category":36,"text_question":268,"slug":269},534373,"A popular cell phone family plan provides 1500 minutes. It charges 89.99/month for the first 2 lines and 9.99 for every line after that. Unlimited text messages for all phone lines costs $30.00/month, and Internet costs $10.00/month per phone line. If a family with a $200 monthly budget buys this plan and signs up for unlimited text messaging and Internet on each phone line, how many cell phone lines can they afford? Use an inequality to solve this problem. Graph your solution on the number line and explain the meaning of your graph in a sentence.","a-popular-cell-phone-family-plan-provides-1500-minutes-it-charges-89-99-month-for-the-first-2-lines-and-9-99-for-every-line-after-that-unlimited-text-messages-for-all-phone-lines-costs-30-00-month",{"id":271,"category":36,"text_question":272,"slug":273},534377,"The business college computing center wants to determine the proportion of business students who have personal computers (PC's) at home. If the proportion is greater than 35%, then the lab will modify a proposed enlargement of its facilities. Suppose a hypothesis test is conducted and the test statistic is z= 2.6. Find the P-value for this test.","the-business-college-computing-center-wants-to-determine-the-proportion-of-business-students-who-have-personal-computers-pc-s-at-home-if-the-proportion-is-greater-than-35-then-the-lab-will-modify",{"id":275,"category":36,"text_question":276,"slug":277},534413,"15.A newly married couple purchased a home with a $123710 down payment. They financed the remaining balance of the home with a mortgage. Their payments were $15395 at the end of every six months for 23 years and the interest rate was 10.6%, compounded semi-annually. How much did they purchase their home for.\n\nEnter to the nearest cent (two decimals). Do not use $ signs or commas in the answer.","15-a-newly-married-couple-purchased-a-home-with-a-123710-down-payment-they-financed-the-remaining-balance-of-the-home-with-a-mortgage-their-payments-were-15395-at-the-end-of-every-six-months-for-2",{"id":279,"category":36,"text_question":280,"slug":281},534417,"Let X be a discrete random variable such that E(X)=3 and V(X)=5. Let 𝑌 = 2𝑋^2 − 3𝑋. Determine E(Y).","let-x-be-a-discrete-random-variable-such-that-e-x-3-and-v-x-5-let-y-2x-2-3x-determine-e-y",{"id":283,"category":36,"text_question":284,"slug":285},534428,"Two particles of electrical charges Q1=3.8×10-⁶C and q,=4.4×10-⁶C are separated in vacuum by a distance of 4.0.10-⁸ m. Since K=9.0.10⁹ N.m²/C², the intensity of the interaction force between them, in newtons, is?","two-particles-of-electrical-charges-q1-3-8-10-c-and-q-4-4-10-c-are-separated-in-vacuum-by-a-distance-of-4-0-10-m-since-k-9-0-10-n-m-c-the-intensity-of-the-interaction-force-between-them-in",{"id":287,"category":36,"text_question":288,"slug":289},534441,"A buyer purchased a North Carolina home for $475,250. The seller allowed the buyer to assume his first small mortgage with a loan balance of $110,000. How much is the excise tax paid in the transaction?\n$951\n$729.50\n$950.50\n$221\nnone of the above","a-buyer-purchased-a-north-carolina-home-for-475-250-the-seller-allowed-the-buyer-to-assume-his-first-small-mortgage-with-a-loan-balance-of-110-000-how-much-is-the-excise-tax-paid-in-the-transactio",{"id":291,"category":36,"text_question":292,"slug":293},534576,"Solve the following system of equations using substitution.\ny=-4x- 11.\n3x+7y=-2","solve-the-following-system-of-equations-using-substitution-y-4x-11-3x-7y-2",{"id":295,"category":36,"text_question":296,"slug":297},534652,"A small box measures 10 in. by 4 in. by 6 in. high. Find the volume of the box.","a-small-box-measures-10-in-by-4-in-by-6-in-high-find-the-volume-of-the-box",{"id":299,"category":36,"text_question":300,"slug":301},534674,"Solve the system of equations by the addition method.\n\n0.01x-0.08y=-0.1\n0.2x+0.6y=0.2","solve-the-system-of-equations-by-the-addition-method-0-01x-0-08y-0-1-0-2x-0-6y-0-2",{"data":303},[304,313,321,331,343,352,361,370,382,390,401,409,420,429,436,443,451,462,470,480],{"id":305,"name":306,"photo":307,"certificates":308,"diplomas":310,"biography":311,"rating":312,"total_answer":206},19,"Seamus","https://api.math-master.org/img/experts/19/19.webp",[309],"https://api.math-master.org/img/experts/19/certificates/certif.webp",[],"I had very much interest from the Starting with simple addition and multiplication, my math journey led me to tackle more challenging problems. What I love most about math is how it's like a puzzle-solving adventure. It helps us understand things around us by finding patterns and connections. It's like a secret code that unveils the mysteries of the world, and I enjoy how each solution feels like a small victory. Math is like a playground for our imagination, where we can come up with creative ways to solve problems and learn new things. It's all about exploring, discovering, and having fun along the way.",4.9,{"id":314,"name":315,"photo":316,"certificates":317,"diplomas":318,"biography":320,"rating":312,"total_answer":138},18,"Nash","https://api.math-master.org/img/experts/18/18.webp",[],[319],"https://api.math-master.org/img/experts/18/diplomas/diploma.webp","My interest in mathematics boomed in my school days which led me to choose mathematics as my core subject in college and later pursue Engineering in University. I have been working at MathMaster for 10 months now and the fact that I like solving math tasks makes me interested in this field. I can work part-time at any available shift for the day and not get exhausted. Moreover solving math tasks keep my brain sharp and focused.\n\n",{"id":322,"name":323,"photo":324,"certificates":325,"diplomas":327,"biography":329,"rating":312,"total_answer":330},15,"Miles","https://api.math-master.org/img/experts/15/15.webp",[326],"https://api.math-master.org/img/experts/15/certificates/2fd3f3a1-34c4-417d-a85a-1459c7fc0356.webp",[328],"https://api.math-master.org/img/experts/15/diplomas/diploma.webp","My name is Ali Haider and I hold a bachelor’s degree in Electrical Engineering and I am also certified Math Expert. I have been teaching Mathematics and Physics for the past 6 years at both the high school and college level. My pass rate for Mathematics is 100%.\nIn addition to teaching, I love playing badminton and video games.\n",108,{"id":13,"name":332,"photo":333,"certificates":334,"diplomas":338,"biography":340,"rating":341,"total_answer":342},"Timmothy","https://api.math-master.org/img/experts/3/3.webp",[335,336,337],"https://api.math-master.org/img/experts/3/certificates/14d1b957-4891-43e9-b5bc-07fdd4d9af21.webp","https://api.math-master.org/img/experts/3/certificates/27b13919-712a-4712-b27a-e09fc16c774f.webp","https://api.math-master.org/img/experts/3/certificates/b70a3a4e-33de-47d7-b0a5-9fbefe397734.webp",[339],"https://api.math-master.org/img/experts/3/diplomas/19c9addf-312a-4437-af41-e89bc3843209 (1).webp","Hello, I'm Jamil Omar, and I want to take you on a journey through my experience with math, from school to university.\nEarly School Years:\nHigh school marked the beginning of my mathematical journey, a journey that started with skepticism but transformed into something wonderful. At first, I wasn't a fan of math, until a special teacher changed everything. This teacher showed me the beauty of math, and suddenly, everything clicked.\nBecause of this teacher, I started enjoying math and even joined math competitions during my high school days. Surprisingly, I excelled and became a champion at both the provincial and regional levels. These wins boosted my confidence and made me realize that I could excel in math.\nThese victories also gave me the confidence to pursue an engineering degree in college. Thanks to that teacher and my successes in math, I knew I could conquer new challenges and make the most of my education journey.\nUniversity Journey:\nMy strong grasp of mathematics inspired me to pursue a bachelor's degree in Petroleum Engineering at Palawan State University. Confident in my mathematical foundation, I embarked on this academic journey.\nThroughout the course, I discovered that my solid background in mathematics was a true asset. It enabled me to better comprehend the intricate topics within Petroleum Engineering and successfully complete the program.\nMy enthusiasm for engineering led me to actively engage in quiz bee contests, where sharp mathematical skills are crucial for success. Our team's victory in these contests opened doors to a national quiz bee in our country. This experience not only honed my mathematical abilities but also granted us the privilege to participate in seminars related to our field of study.\nOne of my achievements was securing the top rank in our departmental exam for algebra. During my first year in college, I outperformed all other first-year students across the university. This accomplishment underscores the significant impact of mathematics on my academic journey.\nStudying math has proven to be invaluable, not just in terms of academic achievements but also in enhancing my analytical skills and enabling me to earn a degree in Petroleum Engineering.",4.8,96,{"id":344,"name":345,"photo":346,"certificates":347,"diplomas":348,"biography":350,"rating":341,"total_answer":351},22,"Hank","https://api.math-master.org/img/experts/22/22.webp",[],[349],"https://api.math-master.org/img/experts/22/diplomas/12.webp","When I was in elementary and middle school, I did not feel the importance and beauty of mathematics, but when I joined the secondary stage, I began to feel the beauty and importance of mathematics, and I excelled in mathematics. I joined the College of Education, Department of Mathematics, and now I am a mathematics teacher.\nThen after that, I got a master's degree and a doctorate in curricula and methods of teaching mathematics.",100,{"id":353,"name":354,"photo":355,"certificates":356,"diplomas":357,"biography":359,"rating":341,"total_answer":360},35,"Hester","https://api.math-master.org/img/experts/35/35.webp",[],[358],"https://api.math-master.org/img/experts/35/diplomas/03849782-84d8-46fb-b7ed-7e3204335b13.webp","Mathematics has been my favourite subject since childhood. I studied mathematics in secondary education and secured full marks in this subject which gave me great joy and boost as i was attached to this subject. Then I opted Pre-Engineering in my higher secondary education which was mainly about the mathematics. Then I got admission in Engineering University which is ranked as Number 1 Engineering University of my country and studied different subjects of mathematics there on graduate level. I really enjoy while solving the maths questions and get too much involved in the basics and different branches of maths as it is referred as mother of sciences and I am emotionally attached to this subject.\r\n\r",115,{"id":362,"name":363,"photo":364,"certificates":365,"diplomas":367,"biography":369,"rating":341,"total_answer":330},14,"Brice","https://api.math-master.org/img/experts/14/14.webp",[366],"https://api.math-master.org/img/experts/14/certificates/a53728ab-4dc7-40ba-8d75-9938dd4881d9.webp",[368],"https://api.math-master.org/img/experts/14/diplomas/240b3c4c-3403-49ef-aab8-6caa8f694f93.webp","Muhammad Farooq here. I have done Bachelor's in Electrical Engineering and in my institute mode of studying was in English. Though I don not posses and certificate of diploma in English as it costs much here in Pakistan and I was not required to submit this certificate anywhere, though my courses in English as well mathematics are mentioned in my School as well collage certificate so you can check my record in these certificates",{"id":371,"name":372,"photo":373,"certificates":374,"diplomas":377,"biography":379,"rating":380,"total_answer":381},28,"Esmeralda","https://api.math-master.org/img/experts/28/28.webp",[375,376],"https://api.math-master.org/img/experts/28/certificates/883 C.webp","https://api.math-master.org/img/experts/28/certificates/883 CC.webp",[378],"https://api.math-master.org/img/experts/28/diplomas/883 D.webp","From an early age, I've been captivated by the enchanting world of mathematics. It all began during my childhood when the mere sight of numbers and equations ignited a spark within me. This fascination evolved into a lifelong passion, shaping my journey into becoming a dedicated math enthusiast.\r\n\r\nThroughout my formative years in both primary and secondary school, I eagerly participated in various math contests, emerging victorious and accumulating a treasure trove of accolades that celebrated my mathematical prowess. As I embarked on my college journey, I chose a bachelor's degree program that seamlessly intertwined mathematics with computational problem-solving.\r\n\r\nStepping into the realm of teaching was a natural progression for me. Armed with a deep reservoir of mathematical knowledge and an innate ability to explain complex concepts in simple terms, I found myself guiding college entrance exam reviewees through the intricacies of algebra, geometry, trigonometry, and pre-calculus. In the interstices of my journey, I also dabbled in the realm of freelance mathematics. With the luxury of time, I engaged in tackling a diverse array of mathematical problems and coursework. This not only honed my skills further but also cemented my belief that mathematics, while a field of study, is also a boundless realm of exploration and creativity.",4.7,98,{"id":383,"name":384,"photo":385,"certificates":386,"diplomas":387,"biography":388,"rating":380,"total_answer":389},12,"Madelyn","https://api.math-master.org/img/experts/12/12.webp",[],[],"i am kishan kushavaha from india. i am completed my primary education from jawahar Navodaya vidhyalaya Rath hamirpur utter pradesh India. Currently I am pursuing Btech degree in computer science and engineering. Math my favorite subject i am feel very happyness solving the math problems.",86,{"id":391,"name":392,"photo":393,"certificates":394,"diplomas":398,"biography":399,"rating":380,"total_answer":400},13,"Maude","https://api.math-master.org/img/experts/13/13.webp",[395,396,397],"https://api.math-master.org/img/experts/13/certificates/664eea5b-1c59-4243-bed8-3a2ffba05ef7.webp","https://api.math-master.org/img/experts/13/certificates/bccb93ad-6ae1-4c1f-bc57-5c2fc5c479b5.webp","https://api.math-master.org/img/experts/13/certificates/cd4c4772-1927-4836-ad8e-340300efad86.webp",[],"i always fond of math ,TILL my matriculation to graduation math is my fvrt subject\nwhen i was in 9th class i had a teacher named as Sir Saleem this is now died but he was bestest teacher ever\nHe teach us each and every question of every exercise of my book of mathematics and what i do is write all those questions in my register as it is and practiced a lot this build my basics and keep them strong i am highly obliged to my sir May God give him high place in jannah ..\nso come to story i have digested too much math in my mind that when i gave math final exams of board i was too much confident and just revise definitions included in course and did not do any question by hand because every question is remembered by heart , i just go through by my register and believe me i got 74/75 marks . this was my initiation in math field that go wide and wide in other fields of math.\nMy interest in mathematics grew as it related to algorithms and data. I moved up into the complexity of algebra and geometry from arithmetic's simplicity because I was fascinated by their complexities. Calculus revealed the language of transformation, while number theory and abstract algebra drove me farther into abstraction. My ability to do rigorous and challenging analysis was put to the test, building a link to the very core of patterns and structures.\n\nThe applications of theory were made possible by probability, statistics, and discrete mathematics, demonstrating the usefulness of mathematical techniques. I incorporated mathematical ideas into my digital self as the digital era developed, optimizing, forecasting, and innovating.\n\n\n\nMy story is a never-ending quest that is inextricably linked to the beauty and logic of mathematics.",104,{"id":402,"name":403,"photo":404,"certificates":405,"diplomas":406,"biography":407,"rating":380,"total_answer":408},7,"Cristian","https://api.math-master.org/img/experts/7/7.webp",[],[],"I embarked on my academic journey as a sports enthusiast, prior to enrolling in my undergraduate studies. My enthusiasm for mathematics had always been muted, largely owing to the seemingly endless and purposeless calculations it often involved. However, a transformative experience awaited me during my time in college. It was there that I encountered a teacher who, in our very first class, presented a simple yet profound exercise: to complete the sentence \"Mathematics is _______\" with a single word that encapsulated our immediate perception of the subject.\nI hesitated briefly before filling the blank space with the word \"boring.\" To my surprise, my teacher chose the word \"beautiful.\" This striking contrast sparked my curiosity and set the stage for a remarkable journey of discovery. Guided by her teaching and insights, it took me a span of two to three months to truly grasp the hidden allure of mathematics. What I once considered mere calculations revealed themselves to be the building blocks of exquisite mathematical and geometric forms.\nThis revelation marked the turning point in my relationship with mathematics. I came to realize that nothing in the realm of mathematics is devoid of purpose or significance. Behind every calculation lies a tapestry of captivating patterns and shapes. As I delved deeper into the core concepts, my appreciation for the elegance and depth of mathematics grew exponentially.\nThis newfound passion became the bedrock of my academic pursuits. It paved the way for my entrance into the world of mathematical exploration, ultimately leading me to pursue a PhD in mathematics. The journey has brought me under the mentorship of an esteemed 80-year-old supervisor, a venerable professor renowned for his wisdom, at Jacobs University in Germany.\nLooking back, I can trace the evolution of my perspective from skepticism to fascination, all catalyzed by a single teacher's belief in the beauty that mathematics holds. With each step I take in my academic journey, I am reminded that the seeds of interest and passion can be sown in the most unexpected of places, transforming the trajectory of one's life in ways that are both profound and deeply fulfilling.",112,{"id":410,"name":411,"photo":412,"certificates":413,"diplomas":418,"biography":419,"rating":380,"total_answer":179},16,"\nClarabelle","https://api.math-master.org/img/experts/16/16.webp",[414,415,416,417],"https://api.math-master.org/img/experts/16/certificates/00f2469c-54a4-4a86-a1bd-058b62670da1.webp","https://api.math-master.org/img/experts/16/certificates/82090cc4-0853-4d1d-88d0-3c33cd9dd4e4 (1).webp","https://api.math-master.org/img/experts/16/certificates/82090cc4-0853-4d1d-88d0-3c33cd9dd4e4.webp","https://api.math-master.org/img/experts/16/certificates/f073e340-308d-4d20-ab38-8b2c9598137b.webp",[],"Hi,\nI've loved Mathematics and critical thinking for as long as I can remember. I started tutoring my fellow students in grade 6. I've studied Maths in the Faculty of Science. I started tutoring online 10 years ago. I've taken a lot of courses to help me teach the STEM curriculum better ever since.\nRegards\nRania",{"id":22,"name":421,"photo":422,"certificates":423,"diplomas":426,"biography":428,"rating":380,"total_answer":206},"Jett","https://api.math-master.org/img/experts/6/6.webp",[424,425],"https://api.math-master.org/img/experts/6/certificates/a05d6f13-1579-47ba-9f4d-1379fd868a32.webp","https://api.math-master.org/img/experts/6/certificates/fcc9c87f-66d3-4474-adfd-13fee92c6090.webp",[427],"https://api.math-master.org/img/experts/6/diplomas/ff175d15-b3af-4041-89fc-d9ac937be8d4.webp","Hello,\nI am pleased to introduce myself as a Mathmaster Expert 😊\nI am Engr. Paul Eejay Dy, a Registered Civil Engineer in the Philippines and Mathematics has always been my favorite subject.\nStudying Mathematics is a really difficult but fun experience for me and with the help of really great teachers and professors that helped me on this journey, I overcame all the obstacles that came my way until I achieved my dream profession.\nSo for all Mathmaster users, it is a great pleasure to be part of your journey into this rough but fulfilling journey of studying Mathematics. Study hard, and keep fighting for your dreams 😇🙏",{"id":19,"name":430,"photo":431,"certificates":432,"diplomas":433,"biography":434,"rating":380,"total_answer":435},"Rasheed","https://api.math-master.org/img/experts/5/5.webp",[],[],"Born with an insatiable curiosity, I began my journey with mathematics during my school years. From solving basic arithmetic problems to unraveling complex algebraic equations, I developed a deep passion for the subject. Upon entering university, my learnings diving into advanced topics like abstract algebra, numerical analysis and differential equations.\nIn retrospect, my journey through the realm of mathematics has been an exhilarating adventure of discovery, learning, and growth.\n",105,{"id":437,"name":438,"photo":439,"certificates":440,"diplomas":441,"biography":442,"rating":380,"total_answer":330},34,"Dexter","https://api.math-master.org/img/experts/34/34.webp",[],[],"I had done my schooling in SPHS, one of premier academic institutions in the city which covered all courses upto K-12. Mathematics had always been my forte, love & passion which earned me a good amount of reputation and accreditation at various phases of my career. My penchant in Maths was observed by my professors in school who did not hesitate to shower me the honour of one of the best mathematics students taught by them. I had a straight A in Mathematics and related sciences right from school to university level. After crossing the barrier of high school with elan, I did not hesitate to continue my graduation in Mathematics major from JU which enlists within the top 5 universities in the country. I stood first in my faculty in the final year of graduation with 85% score amounting to GPA 4.0. During the 3 year graduation, I silently earned the recognition of the top Mathematics student of the class in the eyes of my professors and my fellow students who used to crowd around me when they got stuck in any topic of maths. Overall my journey as a student of Mathematics was challenging, rewarding, enriching and a superbly enjoyable phase of my career. ",{"id":10,"name":444,"photo":445,"certificates":446,"diplomas":447,"biography":449,"rating":450,"total_answer":400},"Corbin","https://api.math-master.org/img/experts/2/2.webp",[],[448],"https://api.math-master.org/img/experts/2/diplomas/1ea09f59-7559-4ce7-8bad-50bcc43ea9ed.webp","My names is James Agustin Molina, I am from Philippines. Back when I in high school, I was not the best in Math in my class; but, If I were to compare my math skills to my other subjects, I think Math is my best subject. I always find solving and learning math techniques fun. Because I think my Math is the my best subject and I always fascinated in how things work, I took engineering in my college. In college, I really enjoyed math and thinking of pursuing a teaching career. I also took and passed my licensure exam right after graduation. Currently, I am taking my Masters Degree in Mathematics.",4.6,{"id":452,"name":453,"photo":454,"certificates":455,"diplomas":456,"biography":460,"rating":450,"total_answer":461},29,"Eliseo","https://api.math-master.org/img/experts/29/29.webp",[],[457,458,459],"https://api.math-master.org/img/experts/29/diplomas/458 2.webp","https://api.math-master.org/img/experts/29/diplomas/458 d.webp","https://api.math-master.org/img/experts/29/diplomas/458 dd.webp","In my school days I was not good in Mathematics but there was something which always attracted me towards Mathematics as playing with numbers and doing round of questions of different exercises of workbook was fun for me and as a hobby I inclined to Math's. I never thought of teaching mathematics and making a career out of it but somethings bound to happen.",106,{"id":463,"name":464,"photo":465,"certificates":466,"diplomas":467,"biography":468,"rating":450,"total_answer":469},30,"Bud","https://api.math-master.org/img/experts/30/30.webp",[],[],"I started my mathematics journey when i was in school and i went for a lot of competitions in mathematics back then and after worth, i proceeded into the university to study Electronics and Electrical Engineering and the course of studying it, i came across a lot of mathematics course which grounded me alot in mathematics.",95,{"id":471,"name":472,"photo":473,"certificates":474,"diplomas":475,"biography":478,"rating":450,"total_answer":479},31,"Frederik","https://api.math-master.org/img/experts/31/31.webp",[],[476,477],"https://api.math-master.org/img/experts/31/diplomas/344.webp","https://api.math-master.org/img/experts/31/diplomas/344D.webp","Hi, my name is Nitin,\r\nI started taking an interest in maths, because of my grandfather, who was a maths professor at HNBGU University, initially, I was not that good at maths, but with practice, I improved a lot, and I became pretty good at it when I got excellent marks in class 10th maths, I chose maths as my primary subject I completed my college and graduation, currently I am pursuing my master, I loved teaching maths, I also teach maths in a local school, and I want to become a professor in maths just like my grandfather.",97,{"id":481,"name":482,"photo":483,"certificates":484,"diplomas":486,"biography":488,"rating":450,"total_answer":330},21,"Jon","https://api.math-master.org/img/experts/21/21.webp",[485],"https://api.math-master.org/img/experts/21/certificates/41e2a05c-d94f-4bcc-9258-f2a7fe4c3c02.webp",[487],"https://api.math-master.org/img/experts/21/diplomas/9258cda6-9255-4b34-8806-23628b0a8153.webp","I passed my matric from Govt.City Muslim school Faisalabad and got first division in school by getting 981/1000 marks.\nAfter matric i got admission in Shiblee Colege Faisalabad and got 883/1100 marks in degree of inter with Math.\nNow currently i'm student of BS Math and running my last semester.",{"$sicons":490},{"bxl:facebook-circle":491,"bxl:instagram":495,"mdi:web":497,"la:apple":499,"ph:google-logo-bold":502,"ph:google-logo":504},{"left":492,"top":492,"width":493,"height":493,"rotate":492,"vFlip":184,"hFlip":184,"body":494},0,24,"\u003Cpath fill=\"currentColor\" d=\"M12.001 2.002c-5.522 0-9.999 4.477-9.999 9.999c0 4.99 3.656 9.126 8.437 9.879v-6.988h-2.54v-2.891h2.54V9.798c0-2.508 1.493-3.891 3.776-3.891c1.094 0 2.24.195 2.24.195v2.459h-1.264c-1.24 0-1.628.772-1.628 1.563v1.875h2.771l-.443 2.891h-2.328v6.988C18.344 21.129 22 16.992 22 12.001c0-5.522-4.477-9.999-9.999-9.999\"/>",{"left":492,"top":492,"width":493,"height":493,"rotate":492,"vFlip":184,"hFlip":184,"body":496},"\u003Cpath fill=\"currentColor\" d=\"M11.999 7.377a4.623 4.623 0 1 0 0 9.248a4.623 4.623 0 0 0 0-9.248m0 7.627a3.004 3.004 0 1 1 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