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.

You're welcome! I'm glad I could help. Yes, the solution to this problem involves using calculus to find the minimum cost. The first step is to express all the dimensions and variables in terms of one variable, in this case, the height (H). Then, you find the total cost equation and differentiate it with respect to H to find when the cost is at its minimum. Solving this equation will give you the value of H, which you can then substitute back into the equations to find the corresponding values of the other dimensions.

In this case, the least expensive box has a height of 3 meters, a width of 8 meters, and a length of 3 meters. The cost of this box is also calculated to be $560.