implemented new rules lin residents. A key component of these rules is that residents should work no more than 80 hours per we of weekly hours worked in 2022 by a sample of residents at the Tidelands Medical Center. $Use t Distri\n84 86 84 86 79 82 87 81 84 78 74 86\n\nB. What is the point estimate of the population standard deviation?\nNote: Round your answer to 2 decimal places.\nc. What is the margin of error for a 90% confidence interval estimate?\nNote: Round your answer to 2 decimal places.\nd. Develop a 90% confidence interval for the population mean.\nNote: Round your answers to 2 decimal places.","","B. Calculate the sample standard deviation (\\(s\\$):\u003Cbr>\u003Cbr>1. List the data: \\(84, 86, 84, 86, 79, 82, 87, 81, 84, 78, 74, 86\\).\u003Cbr>\u003Cbr>2. Calculate the sample mean (\\(\\bar{x}\\)): \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\bar{x} = \\frac{84 + 86 + 84 + 86 + 79 + 82 + 87 + 81 + 84 + 78 + 74 + 86}{12} = \\frac{991}{12} = 82.58\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Apply the standard deviation formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s = \\sqrt{\\frac{\\sum (x_i - \\bar{x})^2}{n - 1}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s = \\sqrt{\\frac{(84-82.58)^2 + (86-82.58)^2 + \\ldots + (86-82.58)^2}{11}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s = \\sqrt{\\frac{1.98 + 11.78 + 1.98 + 11.78 + 13.04 + 0.3364 + 20.57 + 2.48 + 1.98 + 21.02 + 73.73 + 11.78}{11}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s = \\sqrt{\\frac{40.54 + 36.84 + 2.48 + 34.82}{11}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s = \\sqrt{\\frac{151.46}{11}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s=\\sqrt{15.5379}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s\\approx3.94\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>C. Margin of Error:\u003Cbr>\u003Cbr>1. Determine the critical value for 90% confidence with \\( n = 12 \\) which results in degrees of freedom \\( df = 11 \\). Use the t-distribution table.\u003Cbr>\u003Cbr>2. Critical value (\\(t^*\\)) for 90% confidence interval with \\(df = 11\\) is approximately 1.796.\u003Cbr>\u003Cbr>3. Calculate the margin of error:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>E=t^*\\times\\frac{s}{\\sqrt{n}}=1.796\\times\\frac{3.94}{\\sqrt{12}}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>E=1.796\\times1.137\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>E\\approx2.04\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>D. Develop a 90% Confidence Interval:\u003Cbr>\u003Cbr>1. Use the point estimate, margin of error, and sample mean:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\bar{x}\\pm E=82.58\\pm2.04\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Calculate the confidence interval:\u003Cbr>\u003Cbr>Lower limit: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>82.58-2.04=80.54\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Upper limit: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>82.58+2.04=84.62\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Round each bound to 2 decimal places: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(80.54,84.62)\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Answer: 90% Confidence Interval = \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(80.54,84.62)\u003C/math-field>\u003C/math-field>",422,84,"in-2003-the-accreditation-council-for-graduate-medical-education-acgme-implemented-new-rules-lin-residents-a-key-component-of-these-rules-is-that-residents-should-work-no-more-than-80-hours-per-we",{"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},538093,"FIND THE AREA UNDER THE Standard Normal Distribution: To the right of z = - 2.01","To find the area to the right of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z = -2.01\u003C/math-field>\u003C/math-field> on a standard normal distribution:\u003Cbr />\n\u003Cbr />\n1. We need to find the cumulative distribution function (CDF) value for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z = -2.01\u003C/math-field>\u003C/math-field>. \u003Cbr />\n\u003Cbr />\n2. Using the standard normal distribution table or a calculator, we find that the CDF value for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z = -2.01\u003C/math-field>\u003C/math-field> is approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.0222\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Since this value represents the area to the left of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z = -2.01\u003C/math-field>\u003C/math-field>, the area to the right is:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 - 0.0222 = 0.9778\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the area under the standard normal distribution curve to the right of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>z = -2.01\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.9778\u003C/math-field>\u003C/math-field>.",878,176,"find-the-area-under-the-standard-normal-distribution-to-the-right-of-z-2-01",{"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},538092,"2²","The expression $2^2$ represents 2 raised to the power of 2, which is $2 \\times 2 = 4$. Therefore, the answer is 4.",898,180,"2",{"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},538090,"The ratio of Adam’s weight to John’s weight is 6:5. If Adam weighs 48 KG, find John’s weight.","Let Adam's weight be represented as \\( A \\) and John's weight as \\( J \\). \u003Cbr />\n\u003Cbr />\nGiven the ratio is 6:5, we have:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{A}{J} = \\frac{6}{5} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nWe know Adam's weight \\( A = 48 \\, \\text{KG} \\).\u003Cbr />\n\u003Cbr />\nSo substitute \\( A \\) in the ratio:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{48}{J} = \\frac{6}{5} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nBy cross-multiplying:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 6J = 5 \\times 48 \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> 6J = 240 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nNow, solve for \\( J \\):\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> J = \\frac{240}{6} \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> J = 40 \\, \\text{KG} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, John's weight is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>40 \\text{ KG}\u003C/math-field>\u003C/math-field>.",591,118,"the-ratio-of-adam-s-weight-to-john-s-weight-is-6-5-if-adam-weighs-48-kg-find-john-s-weight",{"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},538089,"David cuts a rope 60 m long into two pieces in the ratio 2:3. What is the length of the shorter piece of rope?","1. Let the lengths of the two pieces of rope be represented as $2x$ and $3x$, since they are in the ratio 2:3.\u003Cbr />\n \u003Cbr />\n2. According to the problem, the sum of the lengths of the two pieces is 60 m, so:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2x + 3x = 60 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Combine like terms:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 5x = 60 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Solve for $x$:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = \\frac{60}{5} \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 12 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. The length of the shorter piece of rope is $2x$, so:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2x = 2 \\times 12 \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2x = 24 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Therefore, the length of the shorter piece of rope is:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 24 \\, \\text{m} \u003C/math-field>\u003C/math-field>",1166,233,"david-cuts-a-rope-60-m-long-into-two-pieces-in-the-ratio-2-3-what-is-the-length-of-the-shorter-piece-of-rope",{"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},538088,"Breanne made pineapple drinks by mixing pineapple syrup and water in the ratio 2:7. If she used 4 L of pineapple syrup, how much water did she use?","1. The ratio of pineapple syrup to water is 2:7. This means for every 2 parts of syrup, there are 7 parts of water.\u003Cbr />\n2. Breanne used 4 L of pineapple syrup. Set up the proportion:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{2}{7} = \\frac{4}{x} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n where \\( x \\) is the amount of water used.\u003Cbr />\n\u003Cbr />\n3. Cross-multiply to solve for \\( x \\):\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2x = 7 \\cdot 4 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Simplify:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 2x = 28 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Solve for \\( x \\):\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = \\frac{28}{2} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Calculate:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 14 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n7. Therefore, Breanne used 14 L of water. \u003Cbr />\n\u003Cbr />\nAnswer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14 \\text{ L}\u003C/math-field>\u003C/math-field>",783,157,"breanne-made-pineapple-drinks-by-mixing-pineapple-syrup-and-water-in-the-ratio-2-7-if-she-used-4-l-of-pineapple-syrup-how-much-water-did-she-use",{"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},538087,"y=-2(4)^x+1 +1 describe transformation","Solution:\u003Cbr />\n1. Given function:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = -2(4)^{x+1} + 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Base function:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 4^x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Identify transformations step-by-step:\u003Cbr />\n - **Translation horizontally**: The function has \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(x+1)\u003C/math-field>\u003C/math-field> as the exponent instead of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>. This indicates a horizontal shift to the left by 1 unit.\u003Cbr />\n - **Vertical stretch and reflection**: The coefficient before \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n - **Vertical stretch**: The factor \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field> indicates that the function is stretched vertically by a factor of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n - **Reflection**: The negative sign indicates a reflection across the x-axis.\u003Cbr />\n - **Vertical translation**: The \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>+1\u003C/math-field>\u003C/math-field> outside the function indicates a vertical shift upwards by 1 unit.\u003Cbr />\n\u003Cbr />\n4. Describe the complete transformation:\u003Cbr />\n - The function \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 4^x\u003C/math-field>\u003C/math-field> undergoes the following transformations: a horizontal shift to the left by 1 unit, a vertical stretch by a factor of 2, reflection across the x-axis, and finally a vertical shift upwards by 1 unit.",1255,251,"y-2-4-x-1-1-describe-transformation",{"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},538086,"Add the polynomials g(x)=x3-2x2+3x-1+4x2-x+2","Solution: \u003Cbr />\n1. Write down the given polynomials:\u003Cbr />\n- First polynomial: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>g(x) = x^3 - 2x^2 + 3x - 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Second polynomial: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x^2 - x + 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Align and add the polynomials term by term:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>g(x) = x^3 - 2x^2 + 3x - 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x^2 - x + 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Add the corresponding like terms:\u003Cbr />\n- For \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^3\u003C/math-field>\u003C/math-field> terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^3\u003C/math-field>\u003C/math-field>\u003Cbr />\n- For \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^2\u003C/math-field>\u003C/math-field> terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2x^2 + 4x^2 = 2x^2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- For \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x - x = 2x\u003C/math-field>\u003C/math-field>\u003Cbr />\n- For constant terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-1 + 2 = 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. The resulting polynomial after addition is:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^3 + 2x^2 + 2x + 1\u003C/math-field>\u003C/math-field>",739,148,"add-the-polynomials-g-x-x3-2x2-3x-1-4x2-x-2",{"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},538085,"R=3m. Calculate the volume of the sphere. Round to the nearest tenth if necessary","1. The formula for the volume of a sphere is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{4}{3} \\pi R^3 \u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>2. Substitute the given radius \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> R = 3 \\, \\text{m} \u003C/math-field>\u003C/math-field> into the formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{4}{3} \\pi (3)^3 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Calculate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 3^3 = 27 \u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>4. Thus, the volume becomes:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{4}{3} \\pi \\times 27 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Simplify the expression:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{4 \\times 27}{3} \\pi = 36 \\pi \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>6. Use the approximation \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\pi \\approx 3.1416 \u003C/math-field>\u003C/math-field> :\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V \\approx 36 \\times 3.1416 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>7. Calculate the approximate volume:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>V\\approx113.0973\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>8. Round to the nearest tenth:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V \\approx 113.1 \\, \\text{m}^3 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the volume of the sphere is approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 113.1 \\, \\text{m}^3 \u003C/math-field>\u003C/math-field> .",1203,241,"r-3m-calculate-the-volume-of-the-sphere-round-to-the-nearest-tenth-if-necessary",{"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},538084,"Width of 12 in. Calculate the volume of the sphere. Round to the nearest tenth if necessary","1. Identify the radius of the sphere. Given the width is 12 inches, the diameter is 12 inches. Therefore, the radius is half of the diameter:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> r = \\frac{12}{2} = 6 \\, \\text{in} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Use the formula for the volume of a sphere:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{4}{3} \\pi r^3 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Substitute the radius into the formula:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{4}{3} \\pi (6)^3 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Calculate:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{4}{3} \\pi \\times 216 = \\frac{864}{3} \\pi = 288 \\pi \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Approximate using \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\pi \\approx 3.1416 \u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V \\approx 288 \\times 3.1416 = 904.8 \\, \\text{in}^3 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. The volume of the sphere, rounded to the nearest tenth, is approximately:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 904.8 \\, \\text{in}^3 \u003C/math-field>\u003C/math-field>",278,56,"width-of-12-in-calculate-the-volume-of-the-sphere-round-to-the-nearest-tenth-if-necessary",{"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},538083,"Calculate the volume (to the nearest tenth of a cubic centimeter) of a golf ball whose diameter is 4.267cm","1. The formula for the volume of a sphere is given by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>V = \\frac{4}{3} \\pi r^3\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>2. The diameter of the golf ball is given as 4.267 cm, so the radius is half of that: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>r = \\frac{4.267}{2} = 2.1335 \\, \\text{cm}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>3. Substitute the radius into the volume formula: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>V = \\frac{4}{3} \\pi (2.1335)^3\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>4. Calculate the cube of the radius: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(2.1335)^3 = 9.707432537375\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>5. Substitute this back into the formula: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>V=\\frac{4}{3}\\pi\\times9.707432537375\\approx40.7\\,\\text{cm}^3\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>6. The volume of the golf ball is approximately \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>40.7\\,\\text{cm}^3\u003C/math-field>\u003C/math-field> .",1440,288,"calculate-the-volume-to-the-nearest-tenth-of-a-cubic-centimeter-of-a-golf-ball-whose-diameter-is-4-267cm",{"id":114,"category":36,"text_question":115,"photo_question":38,"text_answer":116,"step_text_answer":8,"step_photo_answer":8,"views":117,"likes":41,"slug":118},538082,"Find the length of each base edge (to the nearest tenth of a meter) of the 24m tall glass square pyramids of the Muttart Conservatory in Alberta, Canada, if each contains 5280m^3 of space","1. Volume V of a square pyramid is given by the formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>V = \\frac{1}{3} B h\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>where B is the area of the base and h is the height of the pyramid.\u003Cbr>\u003Cbr>2. Given that the height h = 24 m and the volume V = 5280 m^3.\u003Cbr>\u003Cbr>3. The base is square, so if the side length of the base is s, then:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>B = s^2\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Substituting into the volume formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5280 = \\frac{1}{3} s^2 \\times 24\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Simplify and solve for s^2:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5280 = 8 s^2\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s^2 = \\frac{5280}{8} = 660\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>6. Solve for s:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s = \\sqrt{660} \\approx 25.7\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>7. To find the length of each base edge to the nearest tenth of a meter, compute:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>s \\approx 25.7 \\, \\text{m}\u003C/math-field>\u003C/math-field>",418,"find-the-length-of-each-base-edge-to-the-nearest-tenth-of-a-meter-of-the-24m-tall-glass-square-pyramids-of-the-muttart-conservatory-in-alberta-canada-if-each-contains-5280m-3-of-space",{"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},538081,"An observer is 150 meters away\n distance of a hot air balloon online\n straight line at ground level. From your position,\n measures an elevation angle of 40° up to\n the base of the balloon. At what height is\n find the hot air balloon?","Solution:\u003Cbr />\n1. Dado:\u003Cbr />\n- Distancia horizontal desde el observador hasta la base del globo: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>d = 150 \\ m\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Ángulo de elevación: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\theta = 40^{\\circ}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Usamos la función tangente para encontrar la altura \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>h\u003C/math-field>\u003C/math-field> del globo aerostático. La tangente de un ángulo en un triángulo rectángulo es la razón entre la altura y la distancia horizontal:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\tan(\\theta) = \\frac{h}{d}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Sustituimos los valores conocidos en la ecuación:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\tan(40^{\\circ}) = \\frac{h}{150}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Resolvemos para \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>h\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>h = 150 \\times \\tan(40^{\\circ})\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Calculamos el valor numérico:\u003Cbr />\n* Usando una calculadora, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\tan(40^{\\circ}) \\approx 0.8391\u003C/math-field>\u003C/math-field>\u003Cbr />\n* Entonces: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>h \\approx 150 \\times 0.8391 = 125.865 \\ m\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nLa altura del globo aerostático es aproximadamente \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>125.865 \\ m\u003C/math-field>\u003C/math-field>.",667,133,"an-observer-is-150-meters-away-distance-of-a-hot-air-balloon-online-straight-line-at-ground-level-from-your-position-measures-an-elevation-angle-of-40-up-to-the-base-of-the-balloon-at-what-hei",{"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},538080,"A plane ticket has gone up 18%, now costing $4,720. How much did it cost before the increase?","\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Solution:}\u003C/math-field>\u003C/math-field>\u003Cbr />\n1. Define variables:\u003Cbr />\n- Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P\u003C/math-field>\u003C/math-field> be the original price of the plane ticket.\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P\u003C/math-field>\u003C/math-field> increased by 18% means the new price is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P + 0.18P = 1.18P\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n2. Set up the equation based on the problem statement:\u003Cbr />\n- The new price \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 4,720\u003C/math-field>\u003C/math-field>.\u003Cbr />\n- Therefore, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1.18P = 4,720\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P\u003C/math-field>\u003C/math-field>:\u003Cbr />\n- Divide both sides by 1.18 to isolate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P\u003C/math-field>\u003C/math-field>.\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P = \\frac{4,720}{1.18}\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n4. Calculate:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P \\approx 4,000\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>\\text{Answer:}\u003C/math-field>\u003C/math-field>\u003Cbr />\n- The original price of the plane ticket was approximately USD 4,000.",726,145,"a-plane-ticket-has-gone-up-18-now-costing-4-720-how-much-did-it-cost-before-the-increase",{"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},538078,"H=8mm, r=2mm. Calculate the volume of the cone round to the nearest tenth if necessary","1. Use the formula for the volume of a cone: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{1}{3} \\pi r^2 H \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Substitute the given values: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> H = 8 \\, \\text{mm}, \\, r = 2 \\, \\text{mm} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{1}{3} \\pi (2)^2 (8) \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Calculate \\( (2)^2 \\):\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> (2)^2 = 4 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Substitute and compute:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{1}{3} \\pi (4)(8) \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{1}{3} \\pi (32) \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Calculate the product: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{32}{3} \\pi \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>6. Calculate:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>V\\approx33.51032\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>7. Round to the nearest tenth:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V \\approx 33.5 \\, \\text{mm}^3 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>This is the answer: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V \\approx 33.5 \\, \\text{mm}^3 \u003C/math-field>\u003C/math-field>",631,126,"h-8mm-r-2mm-calculate-the-volume-of-the-cone-round-to-the-nearest-tenth-if-necessary",{"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},538076,"Dividing 218 or 172 by the natural number n, you get a remainder of 11. Dividing n by 11, you get a remainder equal to:","** \u003Cbr>\u003Cbr>1. Since dividing 218 by n gives a remainder of 11, 218 - 11 = 207 is divisible by n : \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>207\\equiv0\\pmod{n}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Similarly, dividing 172 by n gives a remainder of 11, so 172 - 11 = 161 is divisible by n :\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>161\\equiv0\\pmod{n}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. n must be a common divisor of 207 and 161. Find the greatest common divisor of 207 and 161:\u003Cbr>\u003Cbr>- First, find the difference: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 207 - 161 = 46 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>- Find the prime factorization of 46:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 46 = 2 \\times 23 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>- Prime factorization of 161:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 161 = 7 \\times 23 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>- Common factor is 23.\u003Cbr>\u003Cbr>4. Therefore, the possible value of n should be 23 (since other divisions have factors that don't divide both). Now, divide n = 23 by 11:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 23 \\div 11 = 2 \\, \\text{R} \\, 1 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Thus, the remainder of dividing n by 11 is 1\u003Cbr>\u003Cbr>",1233,247,"dividing-218-or-172-by-the-natural-number-n-you-get-a-remainder-of-11-dividing-n-by-11-you-get-a-remainder-equal-to",{"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},538074,"R=24 inches\nCalculate the surface area of the sphere","1. The formula to calculate the surface area of a sphere is given by: \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>2. Substitute the value of the radius \\( R = 24 \\) inches into the formula: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> A = 4 \\pi (24)^2 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Calculate the square of the radius:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> (24)^2 = 576 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Multiply by 4:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 4 \\times 576 = 2304 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. The surface area is:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>A=2304\\pi=7238.23\u003C/math-field>\u003C/math-field> square inches \u003Cbr>\u003Cbr>Therefore, the surface area of the sphere is 7238.23 square inches.",923,185,"r-24-inches-calculate-the-surface-area-of-the-sphere",{"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},538073,"Andrés's age is three times Quan's.\n plus wins and both ages add up to 69 years. Nillar\n both ages.","Solution:\u003Cbr />\n1. Define variables:\u003Cbr />\n- Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a\u003C/math-field>\u003C/math-field> be the age of Andrés.\u003Cbr />\n- Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>q\u003C/math-field>\u003C/math-field> be the age of Quan.\u003Cbr />\n\u003Cbr />\n2. Set up the equations based on the problem:\u003Cbr />\n- Andrés is three times as old as Quan: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a = 3q\u003C/math-field>\u003C/math-field>\u003Cbr />\n- The sum of their ages is 69: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a + q = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Substitute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a = 3q\u003C/math-field>\u003C/math-field> into the second equation:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3q + q = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Simplify the equation:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4q = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>q\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>q = \\frac{69}{4}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Compute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>q\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>q = 17.25\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n7. Find \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a\u003C/math-field>\u003C/math-field> using the equation \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a = 3q\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a = 3 \\times 17.25\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n8. Compute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a = 51.75\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore:\u003Cbr />\n- Quan is approximately 17.25 years old.\u003Cbr />\n- Andrés is approximately 51.75 years old.",553,111,"andres-s-age-is-three-times-quan-s-plus-wins-and-both-ages-add-up-to-69-years-nillar-both-ages",{"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},538072,"Andrew's age is three times John's plus nine years, and their ages add up to 69 years. Find both ages.","Solution:\u003Cbr />\n1. Define variables:\u003Cbr />\n- Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> be Juan's age.\u003Cbr />\n- Andrés' age is then \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x + 9\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n2. Set up the equation for the total age:\u003Cbr />\n- Juan's age \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> plus Andrés' age \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x + 9\u003C/math-field>\u003C/math-field> equals 69.\u003Cbr />\n\u003Cbr />\n3. Equation:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x + (3x + 9) = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Simplify and solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>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 + 3x + 9 = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x + 9 = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x = 60\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 15\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Find Andrés' age:\u003Cbr />\n- Substitute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 15\u003C/math-field>\u003C/math-field> into Andrés' age expression:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x + 9 = 3(15) + 9 = 45 + 9 = 54\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Therefore, the ages are:\u003Cbr />\n- Juan: 15 years\u003Cbr />\n- Andrés: 54 years",531,106,"andrew-s-age-is-three-times-john-s-plus-nine-years-and-their-ages-add-up-to-69-years-find-both-ages",{"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},538071,"Solve the following linear equations:\n 1) 5x-3= 3X+7","Solution:\u003Cbr />\n1. Given Equation:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5x - 3 = 3x + 7\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Subtract \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x\u003C/math-field>\u003C/math-field> from both sides to simplify:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5x - 3x - 3 = 7\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Combine like terms:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x - 3 = 7\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Add 3 to both sides to isolate the term with the variable:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2x = 10\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Divide both sides by 2 to solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>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 = 5\u003C/math-field>\u003C/math-field>",1382,276,"solve-the-following-linear-equations-1-5x-3-3x-7",{"first":6,"last":176,"prev":8,"next":10},188,{"current_page":6,"from":6,"last_page":176,"links":178,"path":211,"per_page":212,"to":212,"total":213},[179,182,184,186,188,190,192,195,198,201,204,207,209],{"url":6,"label":180,"active":181},"1",true,{"url":10,"label":56,"active":183},false,{"url":13,"label":185,"active":183},"3",{"url":16,"label":187,"active":183},"4",{"url":19,"label":189,"active":183},"5",{"url":22,"label":191,"active":183},"6",{"url":193,"label":194,"active":183},7,"7",{"url":196,"label":197,"active":183},8,"8",{"url":199,"label":200,"active":183},9,"9",{"url":202,"label":203,"active":183},10,"10",{"url":205,"label":206,"active":183},187,"187",{"url":176,"label":208,"active":183},"188",{"url":10,"label":210,"active":183},"Next »","https://api.math-master.org/api/question",20,3743,{"data":215},{"id":216,"category":36,"slug":217,"text_question":218,"photo_question":8,"text_answer":219,"step_text_answer":8,"step_photo_answer":8,"views":220,"likes":221,"expert":222},535926,"local-extrema-of-y-e-square-root-of-x","Local extrema of y=e^square root of x","To find the local extrema of the function \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = e^{\\sqrt{x}}\u003C/math-field>\u003C/math-field>, we need to find where the derivative is equal to zero.\u003Cbr />\u003Cbr />Let's start by finding the derivative of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field> with respect to \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> \\frac{dy}{dx} = \\frac{d}{dx} (e^{\\sqrt{x}}) \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />We can use the chain rule, which states that if \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>u(x)\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>v(x)\u003C/math-field>\u003C/math-field> are differentiable functions, then the derivative of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>u(v(x))\u003C/math-field>\u003C/math-field> with respect to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>u'(v(x)) \\cdot v'(x)\u003C/math-field>\u003C/math-field>.\u003Cbr />\u003Cbr />Using the chain rule:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{dy}{dx} = e^{\\sqrt{x}} \\cdot \\frac{d}{dx}(\\sqrt{x}) = e^{\\sqrt{x}} \\cdot \\frac{1}{2\\sqrt{x}} \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Now, to find the critical points where the derivative is zero, we set \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{dy}{dx} = 0\u003C/math-field>\u003C/math-field>:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> e^{\\sqrt{x}} \\cdot \\frac{1}{2\\sqrt{x}} = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />This equation has no real solutions for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>, which means there are no critical points where the derivative is zero. Therefore, the function \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = e^{\\sqrt{x}}\u003C/math-field>\u003C/math-field> does not have any local extrema.\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{\\text{Answer: There are no local extrema.}}\u003C/math-field>\u003C/math-field>",830,166,{"id":22,"name":223,"photo":224,"biography":225,"created_at":8,"updated_at":8,"rating":226,"total_answer":227},"Jett","https://api.math-master.org/img/experts/6/6.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 😇🙏",4.7,94,{"data":229},{"questions":230},[231,235,239,243,247,251,255,259,263,267,271,275,279,283,287,291,295,299,303,307],{"id":232,"category":36,"text_question":233,"slug":234},532001,"A normal random variable x has a mean of 50 and a standard deviation of 10. Would it be unusual to see the value x = 0? Explain your answer.","a-normal-random-variable-x-has-a-mean-of-50-and-a-standard-deviation-of-10-would-it-be-unusual-to-see-the-value-x-0-explain-your-answer",{"id":236,"category":36,"text_question":237,"slug":238},532041,"The strength of Kefexin oral suspension is 100 mg/ml.\nNora has been prescribed cefalexin at a dose of 50 mg/kg/day divided in two single doses.\nNora weighs 14 kg.\nHow many milliliters of solution for Nora should be given as a single dose?","the-strength-of-kefexin-oral-suspension-is-100-mg-ml-nora-has-been-prescribed-cefalexin-at-a-dose-of-50-mg-kg-day-divided-in-two-single-doses-nora-weighs-14-kg-how-many-milliliters-of-solution-for",{"id":240,"category":36,"text_question":241,"slug":242},532050,"The time it takes for a person to travel 300 m is 15 minutes. What is their speed in meters per second?","the-time-it-takes-for-a-person-to-travel-300-m-is-15-minutes-what-is-their-speed-in-meters-per-second",{"id":244,"category":36,"text_question":245,"slug":246},532311,"2x-y=5 x-y=4","2x-y-5-x-y-4",{"id":248,"category":36,"text_question":249,"slug":250},533945,"Let I ⊂ R be a bounded and nonempty interval. Show that there are numbers a, b ∈ R with a ≤ b and\n I =[a,b] or I =[a,b) or I =(a,b] or I =(a,b)","let-i-r-be-a-bounded-and-nonempty-interval-show-that-there-are-numbers-a-b-r-with-a-b-and-i-a-b-or-i-a-b-or-i-a-b-or-i-a-b",{"id":252,"category":36,"text_question":253,"slug":254},533995,"Suppose 56% of politicians are lawyers if a random sample of size 564 is selected, what is the probability that the proportion of politicians who are lawyers will differ from the total politicians proportions buy more than 4% round your answer to four decimal places","suppose-56-of-politicians-are-lawyers-if-a-random-sample-of-size-564-is-selected-what-is-the-probability-that-the-proportion-of-politicians-who-are-lawyers-will-differ-from-the-total-politicians-pro",{"id":256,"category":36,"text_question":257,"slug":258},534124,"Substitute a=2 and b=-3 and c=-4 to evaluate 2ac/(-2b^2-a)","substitute-a-2-and-b-3-and-c-4-to-evaluate-2ac-2b-2-a",{"id":260,"category":36,"text_question":261,"slug":262},534174,"find f(x) for f'(x)=3x+7","find-f-x-for-f-x-3x-7",{"id":264,"category":36,"text_question":265,"slug":266},534302,"The function h(t)=-5t^2+20t+60 models the height in meters of a ball t seconds after it’s thrown .\nWhich describe the intercepts and vertex of this function","the-function-h-t-5t-2-20t-60-models-the-height-in-meters-of-a-ball-t-seconds-after-it-s-thrown-which-describe-the-intercepts-and-vertex-of-this-function",{"id":268,"category":36,"text_question":269,"slug":270},534335,"A Smooth Plane is listed for $195.00. Discounts of 12% and 10% are allowed. If the customer pays cash within 30 days, an additional discount of 3% is granted. What is the cost if a carpenter takes advantage of all the discounts offered?","a-smooth-plane-is-listed-for-195-00-discounts-of-12-and-10-are-allowed-if-the-customer-pays-cash-within-30-days-an-additional-discount-of-3-is-granted-what-is-the-cost-if-a-carpenter-takes-adv",{"id":272,"category":36,"text_question":273,"slug":274},534354,"Jasminder has made 55% of the recipes in a particular cookbook. If there are 9 recipes that he has never\nmade, how many recipes does the cookbook contain?","jasminder-has-made-55-of-the-recipes-in-a-particular-cookbook-if-there-are-9-recipes-that-he-has-never-made-how-many-recipes-does-the-cookbook-contain",{"id":276,"category":36,"text_question":277,"slug":278},534386,"A building lot is in the shape of a triangle with a base of 133 feet and a height of 76 feet.\nWhat is it's area in square feet?","a-building-lot-is-in-the-shape-of-a-triangle-with-a-base-of-133-feet-and-a-height-of-76-feet-what-is-it-s-area-in-square-feet",{"id":280,"category":36,"text_question":281,"slug":282},534426,"2X+2=8","2x-2-8",{"id":284,"category":36,"text_question":285,"slug":286},534487,"Consider mixing 150 ml, 0.1M, HCI with 100 ml, 0.2M, KOH solution.\nDetermine the pH of final solution.","consider-mixing-150-ml-0-1m-hci-with-100-ml-0-2m-koh-solution-determine-the-ph-of-final-solution",{"id":288,"category":36,"text_question":289,"slug":290},534491,"A salesperson earns a base salary of $600 per month plus a commission of 10% of the sales she makes. You discover that on average, it takes you an hour and a half to make $100 worth of sales.\n How many hours will you have to work on average each month for your income to be $2000?","a-salesperson-earns-a-base-salary-of-600-per-month-plus-a-commission-of-10-of-the-sales-she-makes-you-discover-that-on-average-it-takes-you-an-hour-and-a-half-to-make-100-worth-of-sales-how-man",{"id":292,"category":36,"text_question":293,"slug":294},534507,"Find the set of points formed by the expression 𝜋\u003C|𝑧−4+2𝑖|\u003C3𝜋.","find-the-set-of-points-formed-by-the-expression-z-4-2i-3",{"id":296,"category":36,"text_question":297,"slug":298},534541,"8. Measurement Jillian measured the distance around a small fish pond to be 27 yards. What would be a good estimate of the distance across the pond: 14 yards, 9 yards, or 7 yards? Explain how you decided.","8-measurement-jillian-measured-the-distance-around-a-small-fish-pond-to-be-27-yards-what-would-be-a-good-estimate-of-the-distance-across-the-pond-14-yards-9-yards-or-7-yards-explain-how-you-deci",{"id":300,"category":36,"text_question":301,"slug":302},534543,"1. The cost to transport 250 packages of cement 120 kilometers is $600. What will be the cost to transport 500 packages 300 kilometers?","1-the-cost-to-transport-250-packages-of-cement-120-kilometers-is-600-what-will-be-the-cost-to-transport-500-packages-300-kilometers",{"id":304,"category":36,"text_question":305,"slug":306},534622,"It costs a manufacturer $2,500 to purchase the tools to manufacture a certain homemade item. If the cost for materials and labor is 60¢ per item produced, and if the manufacturer can sell each item for 90¢, find how many items must he produce and sell to make a profit of $2000?","it-costs-a-manufacturer-2-500-to-purchase-the-tools-to-manufacture-a-certain-homemade-item-if-the-cost-for-materials-and-labor-is-60-per-item-produced-and-if-the-manufacturer-can-sell-each-item-fo",{"id":308,"category":36,"text_question":309,"slug":310},534692,"A plant found at the bottom of a lake doubles in size every 10 days. Yeah\n It is known that in 300 days it has covered the entire lake, indicate how many days it will take to cover\n the entire lake four similar plants.","a-plant-found-at-the-bottom-of-a-lake-doubles-in-size-every-10-days-yeah-it-is-known-that-in-300-days-it-has-covered-the-entire-lake-indicate-how-many-days-it-will-take-to-cover-the-entire-lake-fo",{"data":312},{"questions":313},[314,315,319,323,327,331,335,339,343,347,351,355,359,363,367,371,375,379,383,387],{"id":232,"category":36,"text_question":233,"slug":234},{"id":316,"category":36,"text_question":317,"slug":318},532023,"find the value of the tangent if it is known that the cos@= 1 2 and the sine is negative. must perform procedures.","find-the-value-of-the-tangent-if-it-is-known-that-the-cos-1-2-and-the-sine-is-negative-must-perform-procedures",{"id":320,"category":36,"text_question":321,"slug":322},532056,"5 squirrels were found to have an average weight of 9.3 ounces with a sample standard deviation is 1.1. Find the 95% confidence interval of the true mean weight","5-squirrels-were-found-to-have-an-average-weight-of-9-3-ounces-with-a-sample-standard-deviation-is-1-1-find-the-95-confidence-interval-of-the-true-mean-weight",{"id":324,"category":36,"text_question":325,"slug":326},532091,"the value of sin 178°58'","the-value-of-sin-178-58-39",{"id":328,"category":36,"text_question":329,"slug":330},532308,"90 divided by 40","90-divided-by-40",{"id":332,"category":36,"text_question":333,"slug":334},533896,"What payment 7 months from now would be equivalent in value to a $3,300 payment due 23 months from now? The value of money is 2.7% simple interest. Round your answer to 2 decimal places.\nShow all work and how you arrive at the answer..","what-payment-7-months-from-now-would-be-equivalent-in-value-to-a-3-300-payment-due-23-months-from-now-the-value-of-money-is-2-7-simple-interest-round-your-answer-to-2-decimal-places-show-all-work",{"id":336,"category":36,"text_question":337,"slug":338},533901,"For a temperature range between 177 degrees Celsius to 213 degrees Celsius, what is the temperature range in degrees Fahrenheit.","for-a-temperature-range-between-177-degrees-celsius-to-213-degrees-celsius-what-is-the-temperature-range-in-degrees-fahrenheit",{"id":340,"category":36,"text_question":341,"slug":342},533913,"Use the elimination to find the solution to each linear system.\n\nX+y=43\n2x-y=20","use-the-elimination-to-find-the-solution-to-each-linear-system-x-y-43-2x-y-20",{"id":344,"category":36,"text_question":345,"slug":346},533955,"Solve the math problem \r\n400 students are asked if they live in an apartment and have a pet:\r\nApartment: 120\r\nBoth: 30\r\nPet: 90\r\n\r\nThe probability that a randomly selected student not living in an apartment has a pet is","solve-the-math-problem-400-students-are-asked-if-they-live-in-an-apartment-and-have-a-pet-apartment-120-both-30-pet-90-the-probability-that-a-randomly-selected-student-not-living-in-an-apa",{"id":348,"category":36,"text_question":349,"slug":350},533964,"6. Among 100 of products there are 20 rejects. We will randomly select 10 of products. The random variable\r\n\r\n\r\nX indicates the number of rejects among the selected products. Determine its distribution.","6-among-100-of-products-there-are-20-rejects-we-will-randomly-select-10-of-products-the-random-variable-x-indicates-the-number-of-rejects-among-the-selected-products-determine-its-distributio",{"id":352,"category":36,"text_question":353,"slug":354},534100,"Reparameterize the curve r(t)= cos(t)i without (t)j (t)k by the arc length.","reparameterize-the-curve-r-t-cos-t-i-without-t-j-t-k-by-the-arc-length",{"id":356,"category":36,"text_question":357,"slug":358},534106,"Find the equation of the line perpendicular to −5𝑥−3𝑦+5=0 passing through the point (0,−2)","find-the-equation-of-the-line-perpendicular-to-5x-3y-5-0-passing-through-the-point-0-2",{"id":360,"category":36,"text_question":361,"slug":362},534126,"∫ √9x + 1 dx","9x-1-dx",{"id":364,"category":36,"text_question":365,"slug":366},534323,"A function is considered exponential when it has a base with positive values greater than zero and different from one, where the exponent is an unknown.\n\n An important characteristic of exponential functions is that they show rapid growth or decay as an independent variable increases or decreases.\n\n Given the function 25^(x+3)=125, it is calculated that x has the value of","a-function-is-considered-exponential-when-it-has-a-base-with-positive-values-greater-than-zero-and-different-from-one-where-the-exponent-is-an-unknown-an-important-characteristic-of-exponential-fu",{"id":368,"category":36,"text_question":369,"slug":370},534372,"7.57 Online communication. A study suggests that the average college student spends 10 hours per week communicating with others online. You believe that this is an underestimate and decide to collect your own sample for a hypothesis test. You randomly sample 60 students from your dorm and find that on average they spent 13.5 hours a week communicating with others online. A friend of yours, who offers to help you with the hypothesis test, comes up with the following set of hypotheses. Indicate any errors you see.\nH0 :x ̄\u003C10hours HA : x ̄ > 13.5 hours","7-57-online-communication-a-study-suggests-that-the-average-college-student-spends-10-hours-per-week-communicating-with-others-online-you-believe-that-this-is-an-underestimate-and-decide-to-collect",{"id":372,"category":36,"text_question":373,"slug":374},534432,"Calculate the change in internal energy of a gas that receives 16000 J of heat at constant pressure (1.3 atm) expanding from 0.100 m3 to 0.200 m3.\n\n Question 1Answer\n to.\n 7050J\n\n b.\n 2125J\n\n c.\n None of the above\n\n d.\n 2828J\n\n and.\n 10295 J","calculate-the-change-in-internal-energy-of-a-gas-that-receives-16000-j-of-heat-at-constant-pressure-1-3-atm-expanding-from-0-100-m3-to-0-200-m3-question-1answer-to-7050j-b-2125j-c-none",{"id":376,"category":36,"text_question":377,"slug":378},534502,"a) 6x − 5 > x + 20","a-6x-5-x-20",{"id":380,"category":36,"text_question":381,"slug":382},534538,"A 20-year old hopes to retire by age 65. To help with future expenses, they invest $6 500 today at an interest rate of 6.4% compounded annually. At age 65, what is the difference between the exact accumulated value and the approximate accumulated value (using the Rule of 72)?","a-20-year-old-hopes-to-retire-by-age-65-to-help-with-future-expenses-they-invest-6-500-today-at-an-interest-rate-of-6-4-compounded-annually-at-age-65-what-is-the-difference-between-the-exact-acc",{"id":384,"category":36,"text_question":385,"slug":386},534611,"Convert (324)𝑓𝑖𝑣𝑒 into base-ten","convert-324-five-into-base-ten",{"id":388,"category":36,"text_question":389,"slug":390},534696,"In a cheese factory, one pie costs 3800 denars. The fixed ones\ncosts are 1,200,000 denars, and variable costs are 2,500 denars per pie. To encounter:\na) income functions. profit and costs;\nb) the break-even point and profit and loss intervals.","in-a-cheese-factory-one-pie-costs-3800-denars-the-fixed-ones-costs-are-1-200-000-denars-and-variable-costs-are-2-500-denars-per-pie-to-encounter-a-income-functions-profit-and-costs-b-the-brea",{"data":392},[393,397,401],{"id":394,"question":395,"answer":396},104604,"What is the simplified form of (x^3 * x^7) / (x^2)^4?","The answer is x^7. According to the exponent rules, we multiply exponents when raising a power to another power, so (x^2)^4 simplifies to x^8. Then, we divide the numerator exponents by the denominator exponent, giving us x^(3+7-8) = x^7.",{"id":398,"question":399,"answer":400},124633,"Question: Graph the logarithmic function y = log(x) on a coordinate plane. Identify the domain, range, and any asymptotes.","Answer: The logarithmic function y = log(x) has a domain of x > 0, a range of all real numbers, and a vertical asymptote at x = 0. As x approaches ∞, y increases without bound and as x approaches 0⁺, y approaches negative infinity. The graph is a smooth curve that passes through the point (1, 0) and symmetric to the line y = x.",{"id":402,"question":403,"answer":404},145287,"Math question: What is the limit of (x^2 - 4) / (x - 2) as x approaches 2?","Answer: By factoring the numerator, we get (x + 2)(x - 2) / (x - 2). Simplifying, we get x + 2. 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