SOLUTION: a box with a square base has no top. if 64 cm squared material is used, what is the maximum possible volume for the box?

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Question 1151928: a box with a square base has no top. if 64 cm squared material is used, what is the maximum possible volume for the box?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the side of the square base, and let y be the height (depth) of the box.

We are to maximize the volume, which is

V+=+%28x%5E2%29%28y%29

We need that volume formula to be in a single variable. Use the given information about the amount of material used to make the box (bottom and four sides) to solve for y in terms of x and substitute in the volume formula.

x%5E2%2B4xy+=+64 [the total area of the base and four sides is 64 square cm]

4xy+=+64-x%5E2
y+=+%2864-x%5E2%29%2F%284x%29+=+16%2Fx-x%2F4

V+=+%28x%5E2%29%28y%29+=+%28x%5E2%29%2816%2Fx-x%2F4%29+=+16x-x%5E3%2F4

Find the derivative and set it to zero to find the value of x that maximizes the volume.

dV%2Fdx+=+16-3x%5E2%2F4

16-3x%5E2%2F4+=+0
3x%5E2%2F4+=+16
3x%5E2+=+64
x%5E2+=+64%2F3
x+=+8%2Fsqrt%283%29

Determine y for that value of x:



Determine the maximum volume:

V+=+%28x%5E2%29%28y%29+=+%2864%2F3%29%28%284%2F3%29sqrt%283%29%29+=+%28256%2F9%29sqrt%283%29