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
Well, this pizza box sounds like a little origami project! Let's crunch some numbers, shall we?
First, we need to find the size of the square base. We know that the area of the base is 144 in², so the side length of the square base is the square root of 144, which is 12 inches.
Now, we need to figure out the size of the rectangular cardboard that we will use. Since we're cutting out 1-inch squares from each corner and middle section, we're essentially reducing the length and width of the cardboard by 2 inches each.
Let's call the shorter side of the rectangular cardboard "x" inches. When we cut out 1-inch squares from both ends, we're reducing it by 2 inches. So, the actual length after cutting will be x - 2 inches.
Similarly, the longer side of the rectangular cardboard will be y - 2 inches, since we're cutting out 1-inch squares from it as well.
Now, we want the base of the pizza box to be a square with a side length of 12 inches. So, we set up an equation:
(x - 2)(y - 2) = 144
At this point, we have an equation with two variables. We can't solve it directly, but we know that the dimensions of the cardboard must be positive numbers.
So, let me tell you a secret. I don't know the specific dimensions of the cardboard off the top of my head, but you can solve the equation for x and y by substituting different positive values. Just make sure that they're greater than 2, because if x or y were equal to 2 or less, then the pizza box would disappear!
Happy math-ing and pizza-eating!
Let's call the size of the shorter side "x" inches and the size of the longer side "y" inches.
When six 1-inch squares are cut from the corners and middle sections, the dimensions of the base will be reduced by 2 inches in both length and width. So, the length of the base will be (x - 2) inches and the width will be (y - 2) inches.
Since the area of the base is 144 in^2, we can set up the following equation:
(x - 2) * (y - 2) = 144
To solve for x and y, we can rearrange the equation:
xy - 2x - 2y + 4 = 144
Now, let's solve the equation for x and y.
xy - 2x - 2y = 140
(x - 2)(y - 2) = 144
Expanding the equation, we get:
xy - 2x - 2y + 4 = 144
xy - 2x - 2y = 140
(x - 2)(y - 2) = 144
xy - 2x - 2y + 4 = 144
xy - 2x - 2y = 140
Let's try to factor this equation.
(x - 12)(y - 12) = 0
Therefore, either x - 12 = 0 or y - 12 = 0.
If x - 12 = 0, then x = 12.
If y - 12 = 0, then y = 12.
Therefore, either x = 12 or y = 12.
So, the size of the cardboard piece that should be used is either 12 inches for the shorter side or 12 inches for the longer side.
To find the size of the cardboard needed, we need to first determine the dimensions of the square base after the corners and middle sections are cut.
Let's denote the shorter side of the rectangle as 'x' inches and the longer side as 'y' inches.
When we cut a 1-inch square from each corner, the dimensions of the base will reduce by 2 inches in each direction. Therefore, the dimensions of the base after the corners are cut will be 'x-2' inches and 'y-2' inches.
Furthermore, when we cut a 1-inch square from the middle sections, we will end up with two equal squares. So, the dimensions of each square will be 'x-4' inches and 'y-4' inches.
Now let's find the area of the square base:
Area of the square base = Side1 * Side2
Since the base is a square, Side1 = Side2. So the area becomes:
Area of the base = (x-4) * (y-4)
According to the given information, the area of the base should be 144 in^2. Therefore, we have the following equation:
(x-4) * (y-4) = 144
Now, let's solve this equation to find the values of x and y.
We can start by listing all the factors of 144:
1 * 144 = 144
2 * 72 = 144
3 * 48 = 144
4 * 36 = 144
6 * 24 = 144
8 * 18 = 144
9 * 16 = 144
12 * 12 = 144
By examining these factor pairs, we can see that the possible values for (x-4) and (y-4) are:
(x-4) = 9 and (y-4) = 16
(x-4) = 16 and (y-4) = 9
Adding 4 to each side, we get:
x = 13 and y = 20
x = 20 and y = 13
Therefore, the dimensions of the original cardboard needed to create the pizza box are 13 inches by 20 inches, or 20 inches by 13 inches.