\n 3.00 m\n Structures - Fund. Stability and Resistance\n w3 $kN/m$\n AB\n 1.00 m\n P3 $kN$\n w2 $kN/m$\n 4.00 m 2.00 m\n P1 $kN$ P2 $kN$\n P3 $kN$\n RA: 0 0 0 00 22\n P1 = $7 + 0$ kN\n P2 = $5 + 2$ kN\n P3 = $4 + 2$ kN\n w1 = $3 + 0$ kN/m","","1. **Calculate the reaction force at A $vertical force equilibrium$:**\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\Sigma F_y = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> R_A - P1 - P2 - P3 - w1 \\times 2 = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> R_A - 7 - 7 - 6 - 6 = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> R_A = -8 \\, \\text{kN} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. **Calculate the reaction moment at A $taking moments about point A$:**\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\Sigma M_A = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> - M_A + P1 \\times 2 + P2 \\times 5 + P3 \\times 9 + \\frac{w1 \\times 2 \\times 1}{2} = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> - M_A + 7 \\times 2 + 7 \\times 5 + 6 \\times 9 + \\frac{3 \\times 2 \\times 1}{2} = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> - M_A + 14 + 35 + 54 + 3 = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> - M_A + 106 = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> M_A = 18.5 \\, \\text{kN} \\cdot \\text{m} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> R_A = -8 \\, \\text{kN} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> M_A = 18.5 \\, \\text{kN} \\cdot \\text{m} \u003C/math-field>\u003C/math-field>",1297,259,"given-the-cantilevered-isostatic-beam-shown-below-determine-the-support-reactions-and-diagrams-of-dec-and-dmf-efforts-a-2-00-m-w1-kn-m-3-00-m-structures-fund-stability-and-resistance-w3",{"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},536260,"Let's apply what we learned in a practical exercise. 200 students respond\n a test and 10% of them answer question 12 incorrectly.\n Considering that all the students answered question 12, how many\n students answer this question correctly?","1. Calculamos cuántos estudiantes respondieron de manera errónea:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>200 \\times 0.10 = 20\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Restamos los estudiantes que respondieron de manera errónea del total de estudiantes:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>200 - 20 = 180\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{180}\u003C/math-field>\u003C/math-field>",575,115,"let-s-apply-what-we-learned-in-a-practical-exercise-200-students-respond-a-test-and-10-of-them-answer-question-12-incorrectly-considering-that-all-the-students-answered-question-12-how-many-stu",{"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},536259,"$𝟏𝟐$ -8 × $2$ -13 × $𝟏𝟐$ -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>$=12-16-156-5$\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>$=12-16-13\\times 12-5$\n$=12-16-156-5$\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>$=-165$\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>$=12-16-13\\times 12-5$\n$=-165$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",819,164,"12-8-2-13-12-5",{"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},536258,"X - y = 15\n2x + 3y = 29","\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>$x=15+y$\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>$\\begin{bmatrix}2(15+y)+3y=29\\end{bmatrix}$\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>$\\begin{bmatrix}30+5y=29\\end{bmatrix}$\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>$y=-\\frac{1}{5}$\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>$x=15-\\frac{1}{5}$\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>$x=\\frac{74}{5}$\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>$x=\\frac{74}{5},y=-\\frac{1}{5}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",745,149,"x-y-15-2x-3y-29",{"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},536257,"In the following problems find the total area bounded by the graph of the function\n given and the x-axis in the given interval.\n 1.y=x-1;[-1,1]","Step 1: Identify the function and interval: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = x - 1; \\quad [-1, 1]\u003C/math-field>\u003C/math-field>\u003Cbr />\nStep 2: Integrate the function over the interval.\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int_{-1}^{1} $x - 1$ \\, dx\u003C/math-field>\u003C/math-field>\u003Cbr />\nStep 3: Find the antiderivative:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int $x - 1$ \\, dx = \\frac{x^2}{2} - x\u003C/math-field>\u003C/math-field>\u003Cbr />\nStep 4: Evaluate from -1 to 1:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left[ \\frac{x^2}{2} - x \\right]_{-1}^{1} = \\left$\\frac{1^2}{2} - 1 \\right$ - \\left$\\frac{(-1)^2}{2} - (-1$ \\right)\u003C/math-field>\u003C/math-field>\u003Cbr />\nStep 5: Simplify the terms inside the brackets:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left$\\frac{1}{2} - 1 \\right$ - \\left$\\frac{1}{2} + 1 \\right$ = -\\frac{1}{2} - \\frac{3}{2} = -2\u003C/math-field>\u003C/math-field>\u003Cbr />\nStep 6: Take the absolute value of the result to get the total area $since area cannot be negative$:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left| -2 \\right| = 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the answer is:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field>",1173,235,"in-the-following-problems-find-the-total-area-bounded-by-the-graph-of-the-function-given-and-the-x-axis-in-the-given-interval-1-y-x-1-1-1",{"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},536256,"Find the area between the curves\n Y=x³\n Y= x^1/3","To find the area between the curves \\(y = x^{1/3}\\) and \\(y = x^3\\):\u003Cbr />\n\u003Cbr />\n1. Find the points of intersection:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x^{3} = x^{1/3} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x^{9} = x \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x(x^8 - 1) = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x(x - 1)(x + 1)(x^{6} + x^4 + x^2 + 1) = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x= 0, +1, -1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nIntersections are at \\( x = -1, 0, 1 \\).\u003Cbr />\n\u003Cbr />\n2. Compute the integral:\u003Cbr />\n \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = \\int_{-1}^{1} (x^{1/3} - x^3) \\, dx \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nUsing symmetry:\u003Cbr />\n \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 2 \\int_{0}^{1} (x^{1/3} - x^3) \\, dx \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Split into two integrals:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 2 \\left( \\int_{0}^{1} x^{1/3} \\, dx - \\int_{0}^{1} x^3 \\, dx \\right) \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Solve the integrals:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\int_{0}^{1} x^{1/3} \\, dx = \\left[ \\frac{3}{4}x^{4/3} \\right]_{0}^{1} = \\frac{3}{4} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\int_{0}^{1} x^3 \\, dx = \\left[ \\frac{x^4}{4} \\right]_{0}^{1} = \\frac{1}{4} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Compute the difference:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2 \\left( \\frac{3}{4} - \\frac{1}{4} \\right) = 2 \\times \\frac{1}{2}= 1 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Area \\( A = 1 \\)\u003Cbr />\n\u003Cbr />\nFinal answer: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 1 \u003C/math-field>\u003C/math-field>",923,185,"find-the-area-between-the-curves-y-x-y-x-1-3",{"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},536255,"dy needs to bake a 20cm diameter cake.\n\n How big should the surface of the tray be at least to fit the cake?","La superficie mínima requerida es el área del círculo inferior del pastel.\n Área = \u003Cmath-field read-only> \\pi\\left(\\frac{d}{2}\\right)^2=\\pi\\times10^2=314.15cm^2 \u003C/math-field>",1131,226,"dy-needs-to-bake-a-20cm-diameter-cake-how-big-should-the-surface-of-the-tray-be-at-least-to-fit-the-cake",{"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},536254,"3 x 1","3*1 es 3 por 1, lo que equivale a 3.",406,81,"3-x-1",{"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},536253,"write the continuous equation of the line that passes through (1,4) and (1,2)","La pendiente de la recta es: \u003Cmath-field read-only> \\frac{4-2}{1-1}=\\frac{2}{0}=\\infty \u003C/math-field>\n Por lo tanto, la recta es una recta vertical que pasa por (1,2) y (1,4)\n Por lo tanto, la ecuación es x=1",532,106,"write-the-continuous-equation-of-the-line-that-passes-through-1-4-and-1-2",{"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},536252,"Sharp Company manufactures a product for which the following standards have been set: Standard Quantity or Hours Standard Price or Rate Standard Cost Direct materials 3 feet $ 5 per foot $ 15 Direct labor? hours? per hour ? During March, the company purchased direct materials at a cost of $55,650, all of which were used in the production of 3,200 units of product. In addition, 4,900 direct labor- hours were worked on the product during the month. The cost of this labor time was $36,750. The following variances have been computed for the month: Materials quantity variance $ 4,500 U\r\n Labor spending variance $ 1,650 F \r\nLabor efficiency variance $ 800 U \r\nRequired: 1. For direct materials: a. Compute the actual cost per foot of materials for March. b. Compute the price variance and the spending variance. 2. For direct labor: a. Compute the standard direct labor rate per hour. b. Compute the standard hours allowed for the month's production. c. Compute the standard hours allowed per unit of product.","1a. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Actual Cost per Foot} = \\frac{55,650}{9600} = 5.8 \\text{ per foot}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n1b. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Materials Price Variance} = (5.8 - 5) \\times 11,130 = 7,080 U\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n1c. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Materials Spending Variance} = 4,500 U + 7,080 U = 11,580 U\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2a. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Standard Direct Labor Rate per Hour} = \\frac{36,750}{4,900} = 7.5 \\text{ per hour}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2b. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Standard Hours Allowed for the Month's Production} = \\frac{4,900 + 800}{7.5} = 4100 \\text{ hours}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2c. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Standard Hours Allowed per Unit of Product} = \\frac{4,100}{3,200} = 1.28 \\text{ hours per unit}\u003C/math-field>\u003C/math-field>",984,197,"sharp-company-manufactures-a-product-for-which-the-following-standards-have-been-set-standard-quantity-or-hours-standard-price-or-rate-standard-cost-direct-materials-3-feet-5-per-foot-15-direct-l",{"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},536251,"Solve the following separable differential equations.\n\n 1. dy/dx=sin5x","\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>$y=\\int \\sin(5x)dx$\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>$=-\\frac{1}{5}\\cos(5x)+c_{1}$\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>$y\\prime =\\sin(5x)$\n$y=-\\frac{1}{5}\\cos(5x)+c_{1}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1068,214,"solve-the-following-separable-differential-equations-1-dy-dx-sin5x",{"id":114,"category":36,"text_question":115,"photo_question":38,"text_answer":116,"step_text_answer":8,"step_photo_answer":8,"views":117,"likes":97,"slug":118},536250,"Suppose that in the market for good X there are 4000 consumers,\n with an individual demand curve given by: 𝑄\n 𝐷 = 30 − 0.75𝑃.\n There are also 100 producers, each of which has a\n supply curve given by: 𝑄\n 𝑆 = −300 + 20𝑃\n Page 5 of 12\n 2.1.1. Determine the equilibrium price and quantity in this\n Marketplace. Represent graphically.","Dado que existem 4000 consumidores, a curva agregada de procura \\( Q_D \\) é:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q_D = 4000 (30 - 0.75P)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nDado que existem 100 produtores, a curva agregada de oferta \\( Q_S \\) é:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q_S = 100 (-300 + 20P)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nPara encontrar o equilíbrio, igualamos \\( Q_D \\) e \\( Q_S \\):\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4000(30 - 0.75P) = 100(-300 + 20P)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nResolvendo a equação acima:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120000 - 3000P = -30000 + 2000P\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120000 + 30000 = 5000P\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>150000 = 5000P\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P = 30\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nSubstituindo o valor de \\( P \\) em \\( Q_D \\) ou \\( Q_S \\):\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q_D = 4000(30 - 0.75 \\cdot 30)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q_D = 4000(30 - 22.5)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q_D = 4000 \\cdot 7.5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q_D = 30000\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nPortanto, a quantidade de equilíbrio é:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Q = 30000\u003C/math-field>\u003C/math-field>",530,"suppose-that-in-the-market-for-good-x-there-are-4000-consumers-with-an-individual-demand-curve-given-by-q-d-30-0-75p-there-are-also-100-producers-each-of-which-has-a-supply-curve-given-by",{"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},536249,"Walking along Hadrian's Wall, a group of walkers encountered a man who claimed\n Discover a magical well. If a bag of money was put into the well, after counting to five, the well\n doubled the money on it. Stunned, the walkers asked him: “How much would it cost us to use the\n Good?\" The man replied: \"Every time you put a bag in the well and count to five,\n give me €148. Do you accept?\" They accepted. The man put his bag in the well, counted until\n five and took it out. The money had doubled. The walkers took €148 from the bag, paid and\n He asked to repeat the operation. But the third time they did it, after paying €148\n According to the man, the wallet no longer had any money in it! How is this possible?","1. Let the initials be \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0\u003C/math-field>\u003C/math-field>\u003Cbr />\n2. After the first doubling, the money becomes \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2M_0\u003C/math-field>\u003C/math-field>\u003Cbr />\n3. After paying the fee, they have \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2M_0 - 148\u003C/math-field>\u003C/math-field> remaining\u003Cbr />\n4. After the second doubling, they get \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2(2M_0 - 148) = 4M_0 - 296\u003C/math-field>\u003C/math-field>\u003Cbr />\n5. After paying the fee again, they have \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4M_0 - 296 - 148 = 4M_0 - 444\u003C/math-field>\u003C/math-field>\u003Cbr />\n6. After the third doubling, they get \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2(4M_0 - 444) = 8M_0 - 888\u003C/math-field>\u003C/math-field>\u003Cbr />\n7. After paying the fee for the last time, they have \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8M_0 - 888 - 148 = 8M_0 - 1036 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n8. Solving for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8M_0 - 1036 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8M_0 = 1036\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0 = \\frac{1036}{8}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0 = 129.50\u003C/math-field>\u003C/math-field>",1366,273,"walking-along-hadrian-s-wall-a-group-of-walkers-encountered-a-man-who-claimed-discover-a-magical-well-if-a-bag-of-money-was-put-into-the-well-after-counting-to-five-the-well-doubled-the-money-on",{"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},536248,"Use the formula for nCr to evaluate the expression.\n9) 7C4","Answer = The combination formula, denoted as \u003Cmath-field read-only>\\( \\binom{n}{r} \\)\u003C/math-field> or \u003Cmath-field read-only>\\( nCr \\)\u003C/math-field>, is used to calculate the number of ways to choose \\( r \\) items from \\( n \\) items without regard to the order of selection. The formula for \u003Cmath-field read-only>\\( \\binom{n}{r} \\)\u003C/math-field> is given by:\n\n\u003Cmath-field read-only>\\[\\binom{n}{r} = \\frac{n!}{r!(n-r)!}\\]\u003C/math-field>\n\nIn this problem, we need to evaluate \u003Cmath-field read-only>\\( 7C4 \\)\u003C/math-field>. Plugging \\( n = 7 \\) and \\( r = 4 \\) into the formula, we get:\n\n\u003Cmath-field read-only>\\[\\binom{7}{4} = \\frac{7!}{4!(7-4)!} = \\frac{7!}{4! \\cdot 3!}\\]\u003C/math-field>\n\nNow, calculate the factorials:\n\n\u003Cmath-field read-only>\\[7! = 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1\\]\u003C/math-field>\n\u003Cmath-field read-only>\\[4! = 4 \\times 3 \\times 2 \\times 1\\]\u003C/math-field>\n\u003Cmath-field read-only>\\[3! = 3 \\times 2 \\times 1\\]\u003C/math-field>\n\nSubstituting these values into the formula, we get:\n\n\u003Cmath-field read-only>\\[\\binom{7}{4} = \\frac{7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1}{(4 \\times 3 \\times 2 \\times 1) \\times (3 \\times 2 \\times 1)}\\]\u003C/math-field>\n\nSimplify by canceling the common terms in the numerator and the denominator:\n\n\u003Cmath-field read-only>\\[\\binom{7}{4} = \\frac{7 \\times 6 \\times 5}{3 \\times 2 \\times 1}\\]\u003C/math-field>\n\nCalculate the numerator and the denominator:\n\n\u003Cmath-field read-only>\\[7 \\times 6 \\times 5 = 210\\]\u003C/math-field>\n\u003Cmath-field read-only>\\[3 \\times 2 \\times 1 = 6\\]\u003C/math-field>\n\nThus,\n\n\u003Cmath-field read-only>\\[\\binom{7}{4} = \\frac{210}{6} = 35\\]\u003C/math-field>\n\nTherefore, \u003Cmath-field read-only>\\( 7C4 = 35 \\)\u003C/math-field>.",935,187,"use-the-formula-for-ncr-to-evaluate-the-expression-9-7c4",{"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},536247,"From 10 names on a ballot, a committee of 3 will be elected to attend a political national 11) convention. How many different committees are possible?","1. Calculate the number of ways to choose 3 names from 10 (combination formula): \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\binom{10}{3} = \\frac{10!}{3!(10-3)!} \u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Simplify the factorials: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\binom{10}{3} = \\frac{10!}{3! \\cdot 7!} \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Cancel out the common terms in the factorials:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\binom{10}{3} = \\frac{10 \\cdot 9 \\cdot 8}{3 \\cdot 2 \\cdot 1} = 120 \u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Final answer:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\binom{10}{3} = 120 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the number of different committees possible is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 120 \u003C/math-field>\u003C/math-field>.",684,137,"from-10-names-on-a-ballot-a-committee-of-3-will-be-elected-to-attend-a-political-national-11-convention-how-many-different-committees-are-possible",{"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},536245,"Write a whole number, n, that is larger than 5. To keep things interesting, try to pick an n different from what your classmates have already chosen.\nGive an example of a number whose nth root (for your n above) is a whole number.\nExplain how you approached finding your example.","1. Choose a number \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\( n \\)\u003C/math-field>\u003C/math-field>\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n = 6\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. To find a whole number whose 6th root is also a whole number, let’s pick a base whole number \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\( m \\)\u003C/math-field>\u003C/math-field>.\u003Cbr>\u003Cbr>3. Choose \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\( m = 2 \\)\u003C/math-field>\u003C/math-field>.\u003Cbr>\u003Cbr>4. Calculate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\( m^n \\)\u003C/math-field>\u003C/math-field>.\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2^6 = 64\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Therefore, 64 is a number whose 6th root is 2, a whole number.\u003Cbr>\u003Cbr>Answer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>64\u003C/math-field>\u003C/math-field>",733,147,"write-a-whole-number-n-that-is-larger-than-5-to-keep-things-interesting-try-to-pick-an-n-different-from-what-your-classmates-have-already-chosen-give-an-example-of-a-number-whose-nth-root-for-yo",{"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},536244,"A company decided to make some gifts containing three different items (A, B and C). In addition to there not being equal quantities of each item, the packages were made by different people who did not pay attention to the instructions given. In the end, it is known that 49 packages contained only item A, 19 contained only item B and 13 contained only item C. And although 22 packages contained items A and B, 24 contained items A and C and 64 contained items B and C. Only 9 packages contained all three items. What was the initial quantity of each item?","Solução:\n\n Item A = (somente item A) + (item A e C) + (item A e B) + (todos os três itens)\n Item A = 49 + 22 + 24 + 9 = 104\n\n Item B = (Item B apenas) + (Item B e C) + (Item A e B) + (Todos os três itens)\n Item B = 19 + 64 + 22 + 9 = 114\n\n Item C = (Item C apenas) + (Item A e C) + (Item B e C) + (Todos os três itens)\n Item C = 13 + 24 + 64 + 9 = 110",1325,265,"a-company-decided-to-make-some-gifts-containing-three-different-items-a-b-and-c-in-addition-to-there-not-being-equal-quantities-of-each-item-the-packages-were-made-by-different-people-who-did-not",{"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},536242,"(Croatia) Find all integer solutions of the equation 4x + y + 4√ xy − 28√ x − 14√ y + 48 = 0.","First, let \\( \\sqrt{x} = a \\) and \\( \\sqrt{y} = b \\), then \\( x = a^2 \\) and \\( y = b^2 \\). Substituting these into the original equation, we get:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4a^2 + b^2 + 4ab - 28a - 14b + 48 = 0\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>This can be rewritten as:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4a^2 + b^2 + 4ab - 28a - 14b + 48 = 0\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>We notice we need the equation to satisfy \\( a \\) and \\( b \\) being integers. Let us analyze when \\( a \\) and \\( b \\) take on whole number values.\u003Cbr>\u003Cbr>Case: \\( a=2 \\)\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4(2)^2 + b^2 + 4(2)b - 28(2) - 14b + 48 = 0 \\implies 16 + b^2 + 8b - 56 - 14b + 48 = 0\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>We simplify this:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>b^2 - 6b + 8 = 0\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Solving this quadratic equation:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>b = \\frac{6 \\pm \\sqrt{36 - 4 \\cdot 8}}{2} = \\frac{6 \\pm \\sqrt{4}}{2} = \\frac{6 \\pm 2}{2}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, \\( b \\) can be:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>b = 4 \\quad \\text{or} \\quad b = 2\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>This leads us to solutions for:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sqrt{x} = 2 \\quad \\Rightarrow \\quad x = 4\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>And \\( \\sqrt{y} = 4 \\Rightarrow y = 16 \\) or \\(\\sqrt{y} = 2 \\Rightarrow y = 4 \\).\u003Cbr>\u003Cbr>So we have:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(x, y) = (4, 16) \\quad \\text{and} \\quad (4, 4)\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Overall, integer solutions are:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(4, 16) \\quad \\text{and} \\quad (4, 4)\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>The answer is:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(x, y) = (4, 16) \\quad \\text{and} \\quad (4, 4)\u003C/math-field>\u003C/math-field>",453,91,"croatia-find-all-integer-solutions-of-the-equation-4x-y-4-xy-28-x-14-y-48-0",{"id":162,"category":36,"text_question":163,"photo_question":38,"text_answer":164,"step_text_answer":8,"step_photo_answer":8,"views":165,"likes":104,"slug":166},536241,"Calculate the length of the arc of the cycle:\n r(t)=(R(t− sent,R(1−cost)), 0≤t≤2π","1. A fórmula para comprimento de arco de uma curva paramétrica \\(r(t) = (x(t), y(t))\\) é:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> L = \\int_{a}^{b} \\sqrt{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2} \\, dt \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Derivamos as funções \\(x(t)\\) e \\(y(t)\\):\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x(t) = R(t - \\sin{t}) \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> y(t) = R(1 - \\cos{t}) \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{dx}{dt} = R(1 - \\cos{t}) \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{dy}{dt} = R\\sin{t} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Utilizamos as derivadas na fórmula de comprimento de arco:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> L = \\int_{0}^{2\\pi} \\sqrt{(R(1 - \\cos{t}))^2 + (R\\sin{t})^2} \\, dt \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = \\int_{0}^{2\\pi} \\sqrt{R^2(1 - \\cos{t})^2 + R^2\\sin{t}^2} \\, dt \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = R \\int_{0}^{2\\pi} \\sqrt{(1 - \\cos{t})^2 + \\sin{t}^2} \\, dt \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Usamos a identidade trigonométrica \\(1 - \\cos{t} = 2\\sin^2{\\left(\\frac{t}{2}\\right)}\\):\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> (1 - \\cos{t})^2 + \\sin^{2}{t} = (2\\sin^2{\\left(\\frac{t}{2}\\right)})^2 + (\\sin{t})^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 4\\sin^4{\\left(\\frac{t}{2}\\right)} + 4\\sin^2{\\left(\\frac{t}{2}\\right)}\\cos^2{\\left(\\frac{t}{2}\\right)} \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 4\\sin^2{\\left(\\frac{t}{2}\\right)}(\\sin^2{\\left(\\frac{t}{2}\\right)} + \\cos^2{\\left(\\frac{t}{2}\\right)}) \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 4\\sin^2{\\left(\\frac{t}{2}\\right)} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Substituindo de volta na integral:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> L = R \\int_{0}^{2\\pi} \\sqrt{4 \\sin^2{\\left(\\frac{t}{2}\\right)}} \\, dt \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 2R \\int_{0}^{2\\pi} \\left|\\sin{\\left(\\frac{t}{2}\\right)}\\right| \\, dt \u003C/math-field>\u003C/math-field>\u003Cbr />\n Como \\( \\sin{\\left(\\frac{t}{2}\\right)} \\) é positivo no intervalo de \\(0\\) a \\(2\\pi\\), podemos omitir o módulo:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 2R \\int_{0}^{2\\pi} \\sin{\\left(\\frac{t}{2}\\right)} \\, dt \u003C/math-field>\u003C/math-field>\u003Cbr />\n Usando a substituição \\( u = \\frac{t}{2} \\), então \\( du = \\frac{1}{2} dt \\), e os limites mudam para \\( 0 \\leq u \\leq \\pi \\):\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2R \\int_{0}^{\\pi} 2\\sin{u} \\, du \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 4R \\int_{0}^{\\pi} \\sin{u} \\, du \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 4R [-\\cos{u}]_{0}^{\\pi} \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 4R [(-\\cos{\\pi}) - (-\\cos{0})] \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 4R [1 + 1] \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> = 8R \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nPortanto, o comprimento de arco da ciclóide é:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8R\u003C/math-field>\u003C/math-field>",985,"calculate-the-length-of-the-arc-of-the-cycle-r-t-r-t-sent-r-1-cost-0-t-2",{"id":168,"category":36,"text_question":169,"photo_question":38,"text_answer":170,"step_text_answer":8,"step_photo_answer":8,"views":171,"likes":172,"slug":173},536240,"Throw a cube numbered from 1 to 6. What is the probability of the number 7, or a number greater than 4, appearing?","1. Identify the total possible outcomes when rolling a six-sided cube:\u003Cbr />\n \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>6\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Determine the number of favorable outcomes. The cube does not have a number 7, but it does have 5 and 6 which are greater than 4:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Calculate the probability using the formula\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(A) = \\frac{\\text{number of favorable outcomes}}{\\text{total number of possible outcomes}}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cbr />\n4. Substitute the values:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P = \\frac{2}{6} = \\frac{1}{3}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the probability is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{3}\u003C/math-field>\u003C/math-field>.",430,86,"throw-a-cube-numbered-from-1-to-6-what-is-the-probability-of-the-number-7-or-a-number-greater-than-4-appearing",{"first":6,"last":131,"prev":175,"next":176},78,80,{"current_page":178,"from":179,"last_page":131,"links":180,"path":213,"per_page":214,"to":215,"total":216},79,1561,[181,184,186,188,191,194,196,199,201,203,206,209,211],{"url":175,"label":182,"active":183},"« Previous",false,{"url":6,"label":185,"active":183},"1",{"url":10,"label":187,"active":183},"2",{"url":189,"label":190,"active":183},76,"76",{"url":192,"label":193,"active":183},77,"77",{"url":175,"label":195,"active":183},"78",{"url":178,"label":197,"active":198},"79",true,{"url":176,"label":200,"active":183},"80",{"url":90,"label":202,"active":183},"81",{"url":204,"label":205,"active":183},82,"82",{"url":207,"label":208,"active":183},186,"186",{"url":131,"label":210,"active":183},"187",{"url":176,"label":212,"active":183},"Next »","https://api.math-master.org/api/question",20,1580,3737,{"data":218},{"questions":219},[220,224,228,232,236,240,244,248,252,256,260,264,268,272,276,280,284,288,292,296],{"id":221,"category":36,"text_question":222,"slug":223},532026,"Simplify the expression sin³(x)+cos³(x), using trigonometric functions","simplify-the-expression-sin-x-cos-x-using-trigonometric-functions",{"id":225,"category":36,"text_question":226,"slug":227},532038,"10!\r\n-\r\n8! =","10-8",{"id":229,"category":36,"text_question":230,"slug":231},532052,"-6n+5=-13","6n-5-13",{"id":233,"category":36,"text_question":234,"slug":235},532058,"11(4x-9)= -319","11-4x-9-319",{"id":237,"category":36,"text_question":238,"slug":239},532060,"5(4x+3)=75","5-4x-3-75",{"id":241,"category":36,"text_question":242,"slug":243},533882,"Given the vectors: a = (2m – 3n, 4n – m) and b = (2, -3), find the values of m and n that make: a = 5 b.","given-the-vectors-a-2m-3n-4n-m-and-b-2-3-find-the-values-of-m-and-n-that-make-a-5-b",{"id":245,"category":36,"text_question":246,"slug":247},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":249,"category":36,"text_question":250,"slug":251},533916,"math question a bookstore announces a promotion valid for the same purchase as follows: buy a book and get 10% off the total purchase! buy two books and get 20% off your total purchase! buy three or more books and get 30% off your total purchase! Marcelo wanted to buy three books that cost 20.00 each without discount but he decided to buy two books in one day and another purchase with the third book the next day. If he had bought the three books at once he would have saved the following amount.","math-question-a-bookstore-announces-a-promotion-valid-for-the-same-purchase-as-follows-buy-a-book-and-get-10-off-the-total-purchase-buy-two-books-and-get-20-off-your-total-purchase-buy-three-or-m",{"id":253,"category":36,"text_question":254,"slug":255},533921,"A food delivery company charges on average a delivery fee of $5 per order (including food and shipping) and has monthly fixed costs of $600. If the average cost of each meal delivered that is revenue for the company is $10 and the company has a monthly profit of $800, how many orders must they deliver per month?","a-food-delivery-company-charges-on-average-a-delivery-fee-of-5-per-order-including-food-and-shipping-and-has-monthly-fixed-costs-of-600-if-the-average-cost-of-each-meal-delivered-that-is-revenue",{"id":257,"category":36,"text_question":258,"slug":259},533935,"(5-(4-3)*3)-(8+5))","5-4-3-3-8-5",{"id":261,"category":36,"text_question":262,"slug":263},533956,"1 plus 1","1-plus-1",{"id":265,"category":36,"text_question":266,"slug":267},534004,"Suppose 50% of the doctors and hospital are surgeons if a sample of 576 doctors is selected what is the probability that the sample proportion of surgeons will be greater than 55% round your answer to four decimal places","suppose-50-of-the-doctors-and-hospital-are-surgeons-if-a-sample-of-576-doctors-is-selected-what-is-the-probability-that-the-sample-proportion-of-surgeons-will-be-greater-than-55-round-your-answer-to",{"id":269,"category":36,"text_question":270,"slug":271},534060,"-4y-6(2z-4y)-6","4y-6-2z-4y-6",{"id":273,"category":36,"text_question":274,"slug":275},534200,"suppose random variable x follows poisson distribution with expected value 3. what is variance of x?","suppose-random-variable-x-follows-poisson-distribution-with-expected-value-3-what-is-variance-of-x",{"id":277,"category":36,"text_question":278,"slug":279},534233,"The following table shows the frequency of care for some animal species in a center specializing in veterinary dentistry.\n\n\n\n Species %\n Dog 52.8\n Cat 19.2\n Chinchilla 14.4\n Marmoset 6.2\n\n\n Consider that the center only serves 10 animals per week. For a given week, what is the probability that at least two are not dogs?\n\n\n\n ATTENTION: Provide the answer to exactly FOUR decimal places","the-following-table-shows-the-frequency-of-care-for-some-animal-species-in-a-center-specializing-in-veterinary-dentistry-species-dog-52-8-cat-19-2-chinchilla-14-4-marmoset-6-2-consider-t",{"id":281,"category":36,"text_question":282,"slug":283},534422,"The mass of 120 molecules of X2C4 is 9127.2 amu. Identify the unknown atom, X, by finding the atomic mass. The atomic mass of C is 12.01 amu/atom","the-mass-of-120-molecules-of-x2c4-is-9127-2-amu-identify-the-unknown-atom-x-by-finding-the-atomic-mass-the-atomic-mass-of-c-is-12-01-amu-atom",{"id":285,"category":36,"text_question":286,"slug":287},534458,"Given the word WEIRD, determine a four-letter offspring that can be formed with the letters of the word written above","given-the-word-weird-determine-a-four-letter-offspring-that-can-be-formed-with-the-letters-of-the-word-written-above",{"id":289,"category":36,"text_question":290,"slug":291},534500,"The average undergraduate cost per tuition, fees, room, and board for all institutions last year\nwas $26,025. A random sample of 40 institutions of higher learning this year indicated that the\nmean tuition, fees, room, and board for the sample was $27,690, and the population standard\ndeviation is $5492. At the 0.05 level of significance, is there sufficient evidence that the cost has\nincreased? (Remember to follow the steps in hypothesis testing)","the-average-undergraduate-cost-per-tuition-fees-room-and-board-for-all-institutions-last-year-was-26-025-a-random-sample-of-40-institutions-of-higher-learning-this-year-indicated-that-the-mean-tu",{"id":293,"category":36,"text_question":294,"slug":295},534663,"x(squared) -8x=0","x-squared-8x-0",{"id":297,"category":36,"text_question":298,"slug":299},534699,"Construct a set of six pieces of data with​ mean, median, and midrange of 67 and where no two pieces of data are the same.","construct-a-set-of-six-pieces-of-data-with-mean-median-and-midrange-of-67-and-where-no-two-pieces-of-data-are-the-same",{"data":301},[302,312,321,331,343,352,360,369,381,388,399,407,418,427,434,441,450,460,468,477],{"id":303,"name":304,"photo":305,"certificates":306,"diplomas":308,"biography":309,"rating":310,"total_answer":311},19,"Seamus","https://api.math-master.org/img/experts/19/19.webp",[307],"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.",4.9,94,{"id":313,"name":314,"photo":315,"certificates":316,"diplomas":317,"biography":319,"rating":310,"total_answer":320},18,"Nash","https://api.math-master.org/img/experts/18/18.webp",[],[318],"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",85,{"id":322,"name":323,"photo":324,"certificates":325,"diplomas":327,"biography":329,"rating":310,"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":48},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",{"id":361,"name":362,"photo":363,"certificates":364,"diplomas":366,"biography":368,"rating":341,"total_answer":330},14,"Brice","https://api.math-master.org/img/experts/14/14.webp",[365],"https://api.math-master.org/img/experts/14/certificates/a53728ab-4dc7-40ba-8d75-9938dd4881d9.webp",[367],"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":370,"name":371,"photo":372,"certificates":373,"diplomas":376,"biography":378,"rating":379,"total_answer":380},28,"Esmeralda","https://api.math-master.org/img/experts/28/28.webp",[374,375],"https://api.math-master.org/img/experts/28/certificates/883 C.webp","https://api.math-master.org/img/experts/28/certificates/883 CC.webp",[377],"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":382,"name":383,"photo":384,"certificates":385,"diplomas":386,"biography":387,"rating":379,"total_answer":172},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.",{"id":389,"name":390,"photo":391,"certificates":392,"diplomas":396,"biography":397,"rating":379,"total_answer":398},13,"Maude","https://api.math-master.org/img/experts/13/13.webp",[393,394,395],"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":400,"name":401,"photo":402,"certificates":403,"diplomas":404,"biography":405,"rating":379,"total_answer":406},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.",111,{"id":408,"name":409,"photo":410,"certificates":411,"diplomas":416,"biography":417,"rating":379,"total_answer":159},16,"\nClarabelle","https://api.math-master.org/img/experts/16/16.webp",[412,413,414,415],"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":419,"photo":420,"certificates":421,"diplomas":424,"biography":426,"rating":379,"total_answer":311},"Jett","https://api.math-master.org/img/experts/6/6.webp",[422,423],"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",[425],"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":428,"photo":429,"certificates":430,"diplomas":431,"biography":432,"rating":379,"total_answer":433},"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":435,"name":436,"photo":437,"certificates":438,"diplomas":439,"biography":440,"rating":379,"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":442,"photo":443,"certificates":444,"diplomas":445,"biography":447,"rating":448,"total_answer":449},"Corbin","https://api.math-master.org/img/experts/2/2.webp",[],[446],"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.",4.6,103,{"id":451,"name":452,"photo":453,"certificates":454,"diplomas":455,"biography":459,"rating":448,"total_answer":97},29,"Eliseo","https://api.math-master.org/img/experts/29/29.webp",[],[456,457,458],"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.",{"id":461,"name":462,"photo":463,"certificates":464,"diplomas":465,"biography":466,"rating":448,"total_answer":467},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":469,"name":470,"photo":471,"certificates":472,"diplomas":473,"biography":476,"rating":448,"total_answer":342},31,"Frederik","https://api.math-master.org/img/experts/31/31.webp",[],[474,475],"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.",{"id":478,"name":479,"photo":480,"certificates":481,"diplomas":483,"biography":485,"rating":448,"total_answer":486},21,"Jon","https://api.math-master.org/img/experts/21/21.webp",[482],"https://api.math-master.org/img/experts/21/certificates/41e2a05c-d94f-4bcc-9258-f2a7fe4c3c02.webp",[484],"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.",107,{"$sicons":488},{"bxl:facebook-circle":489,"bxl:instagram":493,"mdi:web":495,"la:apple":497,"ph:google-logo-bold":500,"ph:google-logo":503},{"left":490,"top":490,"width":491,"height":491,"rotate":490,"vFlip":183,"hFlip":183,"body":492},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":490,"top":490,"width":491,"height":491,"rotate":490,"vFlip":183,"hFlip":183,"body":494},"\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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