Question

Mrs. Lindsay doesn't have a cat crate and needs to take her cats to the vet. She's
going to make a box out of a 8 foot by 8 foot piece of cardboard. How many feet
should Mrs. Lindsay cut from the corners to have the maximum amount of space
inside the box?

Answer 1

Answer:

The amount to remove from each corner is a square, 4/3 ft by 4/3 ft.

Step-by-step explanation:

The piece of cardboard measures 8 ft by 8 ft.

Each corner will be cut to create the sides of the box.

Let the square cut from each corner measure x by x ft.

The base of the box will measure 8 - 2x by 8 - 2x, and the height of the box will be x.

V = (8 - 2x)^2 * x

V = 64x - 32x^2 + 4x^3

Now we differentiate and set the derivative equal to zero to find a maximum value.

64 - 64x + 12x^2 = 0

3x^2 - 16x + 16 = 0

(3x - 4)(x - 4) = 0

3x - 4 = 0  or  x - 4 = 0

3x = 4   or   x = 4

x = 4/3  or  x = 4

The solution x = 4 must be discarded since 8 - 2x = 8 - 8 = 0, and the sides would have 0 length.

The amount to remove from each corner is a square, 4/3 ft by 4/3 ft.