SOLUTION: A. A box with a square base has a volume of 867 inches cube. Express the surface area of the box as a function of the length x of a side of the base. (Be sure to include the top of

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Question 1097730: A. A box with a square base has a volume of 867 inches cube. Express the surface area of the box as a function of the length x of a side of the base. (Be sure to include the top of the box.)
B. Use the function in part a to find the dimensions of the box with volume 867 inches cube and the smallest possible surface area.
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!

Part A:

Let x be the length of a side of the square base, and let h be the height. The volume is then
V+=+x%5E2h+=+867
which means
h+=+867%2Fx%5E2

The surface area is comprised of the square top and bottom and four sides, each x by h:
S%28x%29+=+2x%5E2+%2B+4xh+=+2x%5E2+%2B+4x%28867%2Fx%5E2%29+=+2x%5E2%2B3468x%5E%28-1%29

Part B:

To minimize the surface area, we need the derivative S'(x) to be zero:
4x-+3468x%5E%28-2%29+=+0
4x%5E3-3468+=+0
x%5E3+=+867

The box with a volume of 867 cubic inches with the smallest surface area is a cube with edge length equal to the cube root of 867.

Note that the calculus answer is what we would expect, since a box with fixed volume and minimum surface area is a cube.