A sheet of cardboard 3 ft by 4 ft will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with the largest volume?

1 Answer
Dec 5, 2017

1.86 ft x 2.86 ft x 0.57 ft

Explanation:

1) Draw out the picture. You will see that the height will be x, the width will be 3-2x, and the length will be 4-2x.
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2) You know the formula for volume will be V = LWH. So, you plug in the numbers you know, creating V = x(3-2x)(4-2x).

3) Simplify by foiling and you get V = 4x^3 - 14x^2 +12x

4) Find the derivative of the equation. You get V' = 12x^2 - 28x +12.

5) Find the critical numbers by seeing where V' = 0 (since V' DNE is not a possible scenario in this problem).
4(3x^2 - 7x +3) = 0
Use the quadratic formula or your calculator to solve for x
x = 0.57, 1.77

6) Check to see which numbers are in the domain.
1.77 is not in the domain since that would make one of our side lengths less than zero. So our only answer is: x = 0.57

7) Plug x back into the equations to get the dimensions
3-2(.57) = 1.86 ft
4-2(.57) = 2.86

So, you know the dimensions are 1.86 ft x 2.86 ft x 0.57 ft.