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 and folding up the sides (see the figure). If the area of the base is to be 144 in^2, what size piece of cardboard should be used?

shorter side in
longer side in

To find the size of the cardboard needed, we need to work backwards from the given information.

Let's assume the shorter side of the rectangular sheet of cardboard is x inches. Since we need to cut out 1-inch squares from the corners and middle sections, the length of the base of the pizza box will be x - 2 inches.

Similarly, the longer side of the rectangular sheet of cardboard will be y inches. The width of the base of the pizza box will be y - 2 inches.

We are given that the area of the base is 144 in^2. The formula for the area of a square is given by A = side^2. Therefore, we have the equation:

(x - 2)(y - 2) = 144

To solve this equation, we need another piece of information. Since we don't have the dimensions of the rectangular sheet of cardboard, it is not possible to determine the exact dimensions.

However, we can approach the problem by substituting different values and checking which values satisfy the equation. We can start by assuming the shorter side (x) to be 10 inches and iterate through various values until we find the one that satisfies the equation.

Assuming x = 10 inches, the equation becomes:

(10 - 2)(y - 2) = 144
8(y - 2) = 144
8y - 16 = 144
8y = 160
y = 20

Therefore, if the shorter side (x) is assumed to be 10 inches, then the longer side (y) would be 20 inches for the area of the base to be 144 in^2.

Please note that this is just an example, and you can try out different values for x to find the corresponding value of y that satisfies the equation.

The final product will have a 12 by 12 base -> 144in^2. With one inch cut out from each corner, you need a (12+1+1) by (12+1+1) base to start.

Since the base is square, all sides are equal.