What will the dimensions of the resulting cardboard box be if the company wants to maximize the volume and they start with a flat piece of square cardboard 20 feet per side, and then cut smaller squares out of each corner and fold up the sides to create the box?

glyperezrl 2022-07-16 Answered
What will the dimensions of the resulting cardboard box be if the company wants to maximize the volume and they start with a flat piece of square cardboard 20 feet per side, and then cut smaller squares out of each corner and fold up the sides to create the box?
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Answers (1)

coolng90qo
Answered 2022-07-17 Author has 14 answers
Suppose that the squares removed from each corner are x feet by x feet each.
When these are folded up they give a box with a height of x feetand a base of 20−2x feet by 20−2x feet for a volume
V = x ( 20 - 2 x ) 2 = 400 x - 80 x 2 + 4 x 3
To find the critical point(s) take the derivative of V, set it to zero, and solve for x.
d V d x = 400 - 160 x + 12 x 2
= 4 ( 3 x - 10 ) ( x - 10 ) = 0
Since x=10 gives a Volume of 0
so the critical point for the Volume that is it's maximum occurs when x = 10 3
The resulting box will be
3 1 3 × 13 1 3 × 13 1 3
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