A pizza box with a square base is to be made from a rectangular sheet of cardboard by cutting six 1-inch squares from the corners and the middle sections folding up the sides. If the area of the base is to be 144 in^2, what size piece of cardboard should be used.

14" x 14", assuming no top cover.

Please explain how you got 14" x 14"

To solve this problem, we need to find the dimensions of the original rectangular sheet of the cardboard that will result in a pizza box with a square base of area 144 in^2.

Let's denote the width of the original rectangular sheet as "w" and its length as "l". We know that when we cut six 1-inch squares from the corners and fold up the sides, the resulting base of the pizza box will be a square with an area of 144 in^2.

When we cut a 1-inch square from each corner of the rectangle, it reduces both the width and length of the rectangle by 2 inches. Therefore, the width of the base of the pizza box is (w - 2) inches and the length is (l - 2) inches.

Since the area of the base is given as 144 in^2, we can create an equation using the formula for the area of a square:

(w - 2) * (l - 2) = 144

This equation represents the dimensions of the base of the pizza box after cutting and folding. We need to solve this equation to find the values of "w" and "l" and, consequently, the size of the rectangular sheet of cardboard.

Let's solve the equation:

w * l - 2w - 2l + 4 = 144

Rearranging the terms:

w * l - 2w - 2l = 140

We can factor out a common factor from the left side:

(w - 2)(l - 2) = 140

Now, we need to find two numbers that, when subtracted by 2, multiply to give 140. The prime factors of 140 are 2 * 2 * 5 * 7.

We can consider various combinations for the factors:

w - 2 = 2
l - 2 = 70

w - 2 = 5
l - 2 = 28

w - 2 = 7
l - 2 = 20

The first combination doesn't make sense since the difference between two sides of the cardboard sheet should not be only 2 inches.

The second combination is valid, and it gives:

w = 5 + 2 = 7 inches
l = 28 + 2 = 30 inches

So, a rectangular sheet of cardboard with dimensions 7 inches by 30 inches should be used to create a pizza box with a square base of area 144 in^2.