# Southern New Hampshire University - 6-3 Module Six Problem Set.pdf

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4/9/21, 9 : 15 AM Southern New Hampshire University - 6-3 Module Six Problem Set Page 1 of 8 [PRINT] MAT-225-X4741 21EW4 Calc I: Single-Variable Calc , 6-3 Module Six Problem Set Dallas Weber, 4/6/21 at 9:12:08 AM EDT Question1: Score 3/6 Find two numbers with difference and whose product is a minimum. Enter your answers in increasing order. First number: Your response Correct response -68 -68 Auto graded Grade: 2/2.0 Second Number: Your response Correct response 68 68 Auto graded Grade: 1/1.0 Show your work and explain, in your own words, how you arrived at your answers. x-y=136 y=x-136 The product of the two numbers is a minimum: xy=x(x-136) f(x)=x^2-136x Set the derivative to 0 for critcal points: f'(x)=2x-136=0 2x=136 -68,68 Ungraded Grade: 0/3.0 Total grade: 1.0 × 2/6 + 1.0 × 1/6 + 0.0 × 3/6 = 33% + 17% + 0% Feedback: The following is Mobius' explanation for a solution to this question. You can use this and other online references as a guide, but your explanation should be in your own words. Let be the smaller number and be the larger number. The product of the two numbers is . If the difference between and is , then . Substitute this formula for into the formula for to get . Differentiate with respect to to obtain . Solve to find the critical numbers. Thus, the critical number is . 136 ! ! " x y P = xy x y 136 y = x + 136 y P P = x ( x + 136) = + 136 x x 2 x = 2 x + 136 dP dx = 0 dP dx 2 x + 136 = 0 2 x = - 136 x = - 68 x = - 68 2
4/9/21, 9 : 15 AM Southern New Hampshire University - 6-3 Module Six Problem Set Page 2 of 8 Since , by the Second Derivative Test, is a local minimum. Therefore, the two numbers are and . Question2: Score 3/3 A farmer wants to fence a rectangular area of square feet next to a river. Find the length and width of the rectangle which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to the river. Length: Your response Correct response 12 12 Auto graded Grade: 1/1.0 feet Width: Your response Correct response 6 6 Auto graded Grade: 1/1.0 feet Total grade: 1.0 × 1/2 + 1.0 × 1/2 = 50% + 50% Feedback: Let be the width and be the length. The amount of fencing required is . If the area required is , then or . Substitute this formula for into the formula for to get . Differentiate with respect to to obtain . Solve to find the critical numbers. Thus, the critical numbers are . Since the width must be greater than zero, we need only consider .
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