A box is to be made where the material for the sides and the lid cost $0.20 per square foot and the cost for the bottom is $0.05per square foot. Find the dimensions of a box with volume 15cubic feet that has minimum cost.

Question

Mathematics

Matilda Wilkerson

2 July, 19:54

A box is to be made where the material for the sides and the lid cost $0.20 per square foot and the cost for the bottom is $0.05per square foot. Find the dimensions of a box with volume 15cubic feet that has minimum cost.

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Caden · Answer 1

Let the width of the base be x and the height of the box be y. The base is a square so its area is x2. Then the volume of the box is "base area times height", so the volume is V=x2y=40ft3.

The area of the base is x2, so the cost of the base is 0.31x2.

The area of each side is xy, so the cost of each side is 0.05xy, so the cost of all four sides is 4*0.05xy=0.2xy.

The area of the top is x2, so the cost of the top is 0.19x2.

So the total cost is C=0.31x2+0.19x2+0.2xy=0.5x2+0.2xy.

Since x2y=40, we have y=40/x2, so C=0.5x2+0.2x*40x2=0.5x2+8x. This we can optimise using derivatives.