Question

A 27-inch by 72-inch piece of cardboard is used to make an open-top box by removing a square from each corner of the cardboard and folding up the flaps on each side. What size square should be cut from each corner to get a box with the maximum volume? Enter the area of the square and do not include any units in your answer.

Answer 1

36

Explanation:

Given:

Length of the cardboard = 27 inches

Width of the cardboard = 72 inches.

Let "x" be side of the square which is cut in each corner.

Now the height of box = "x" inches.

Now the length of the box = 27 - 2x and width = 72 - 2x

Volume (V) = length × width × height

V = (27 - 2x)(72 - 2x)(x)

`V= (1944 -144x -54x + 4x^2)x\\V = (4x^2 - 198x +1944)x\\V = 4x^3 -198x^2 +1944x`

Now let's find the derivative

V' =
`12x^2 - 396x + 1944`

Now set the derivative equal to zero and find the critical points.

`12x^2 - 396x + 1944` = 0

12 (
`x^2 - 33x + 162`) = 0

Solving this equation, we get

x = 6 and x = 27

Here we take x = 6, we ignore x = 27 because we cannot cut 27 inches since the entire length is 27 inches.

So, the area of the square = side × side

= 6 inches × 6 inches

The area of the square = 36 square inches.