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Calculus Optimization?
A box with a square based and closed top must have a volume of 125 cubic inches. Find the dimensions of the box that minimizes the amount of material used. What is the minimum amount of material?
1 Answer
- GuyLv 71 decade agoFavorite Answer
The box that minimizes area for a given volume is a cube. Therefore the side length would be cuberoot(125) = 5 inches, and the surface area would be 6 * 5^2 = 150 square inches.
If you need to do it in a more traditional fashion, then do the following:
x^2 * h = 125
(x is the length of the side of the square base and h is the height of the box)
The surface area is:
A = 2x^2 + 4xh = 2x^2 + 4x(125/x^2)
Solve the equation A'(x) = 0, and you will have your x dimension, which will then determine your height. You will find that h = x.