Question

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?

Answer 1

`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`.

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.