Question

A box with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 8 meters and its volume is 72 cubic meters. If building this box costs $20 per square meter for the base and $10 per square meter for the sides, what is the cost of the least expensive box? What are the dimensions of this least expensive box?

Let H be the height and L the length. The width is W = 8 and the base area is LW . Since the volume must be 72, HLW = 8 HL = 72
Therefore L = 9/H. You can treat H as the single unknown.

Cost = 10(2 HW + 2HL) + 20 WL
= 10(16 H + 18) + 1440/H

For minimum cost, d(Cost)/dH = 0
160 -1440/H^2 = 0
H = 3 meters
W= 8 meters
L = 9/H = 3 meters

Although you listed your subject as just "math". I had to use calculus to do this. I hope you were able to follow the solution.

thanx fo rthe help.