A rectangular storage container with an open top is to have a volume of 10 m3. T

Question

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 $20 per square meter. Material for the sides costs $12 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.) keeping in mind that the volume of an open box with height h and a rectangular base whose length is twice its width w is V = 2w2h, and the cost function of its surface is C = c1(2w2) + c2[2(wh) + 2(2wh)], where c1 is the cost for the base and c2 is the cost for the sides. Find a relationship between w and h, using the fact that the volume is a constant. Rewrite the cost function as a function of one variable. Use calculus to find the minimum possible cost.

Explanation / Answer

cost=($10)LW + $6(2HW + 2HL) W = (1/2)L volume = LWH = 10 m^3 -> so H = 10/LW = 20/(L^2) now we have W and H in terms of L so we can substitute them back into cost to get a function of L only 5L^2 + (360/L) = $cost take derivative (d$cost/dL) to get: -(360/L^2) + 10L = 0 you can now get L = 3.30193 -> back substitute to get: W= 1.65 H= 1.834 And now the cost is: $163.46

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