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 400 in^2, what size piece of cardboard should be used?
To find the size of the cardboard needed, we can break down the problem into steps:
Step 1: Understand the problem and visualize it.
From the given information, we know that a pizza box with a square base is made by cutting six 1-inch squares from the corners and middle sections of a rectangular sheet of cardboard. By folding up the sides, a pizza box is formed. We are asked to find the size of the cardboard needed if the area of the base is 400 in^2.
Step 2: Determine the dimensions of the base after cutting and folding.
Based on the problem description, let's consider the dimensions of the base as follows:
- The side length of the square base after folding is (x - 2)
- The length of the rectangular sheet before cutting is x
- The width of the rectangular sheet before cutting is y
By cutting 1-inch squares from each corner and the middle sections, the resulting dimensions can be calculated as:
- Length of the base = (x - 2) inches
- Width of the base = (y - 2) inches
Step 3: Calculate the area of the base.
The area of a square is found by multiplying its side lengths together. Therefore, the area of the base can be written as:
Area of the base = (x - 2) * (y - 2)
Step 4: Set up and solve the equation.
From the problem statement, we are given that the area of the base is 400 in^2. So, we can set up an equation:
(x - 2) * (y - 2) = 400
Step 5: Solve the equation.
We need to solve the equation to find the values of x and y. After finding the values for x and y, we can determine the size of cardboard required.
Step 6: Determine the answer.
Once we have found the values of x and y, we can calculate the dimensions of the cardboard sheet by adding the 1-inch squares cut from each side to the base dimensions. In this case, the length of the cardboard sheet would be (x + 2) inches, and the width would be (y + 2) inches.
Therefore, to find the size of the cardboard piece required, we need to find the values of x and y that satisfy the equation (x - 2) * (y - 2) = 400.