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9 December, 07:06

A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $5 per square meter. Material for the sides costs $3 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)

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  1. 9 December, 07:15
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    We have to find the cost of materials for the cheapest container. It has a volume of 10 m^3. V = 10 m^3; V = L * W * H; where L is length, W is width and H is height. We know that L = 2 W; 10 = 2 W^2 * H; H = 10 / 2W^2; H = 5 / W^2. Base Area = L * W = 2 W * W = 2 W^2. Side Area = 5 / W^2 * (2 W + 4 W) = 5 / W^2 * 6 W = 30 / W. The Cost: C (W) = 5 * 2 W^2 + 3 * 30 / W = 10 W^2 + 90 / W. We have to find the derivative of the cost. C' (W) = 20 W - 90 / W^2 = (20 W^3 - 90) / W^2. Then: C' (W) = 0; 20 W^3 - 90 = 0; 20 W^3 = 90; W^3 = 4.5; W = 1.651. Finally the cost for the cheapest container is: C (1.651) = 10 * (1.651) ^2 + 90 / 1.651 = 10 * 2.72568 + 54.5157 = 27.2568 + 54.5157 = 81.7725 ($81.77 to the nearest cent). Answer: The cost of materials for the cheapest such container is $81.77.
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