A pizza box top 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. If the area of the base is to 144 in^2, what piece of cardboard should be used.
Please explain the process, thanks in advance.
Well, well, well, looks like we've got ourselves a pizza problem here! Now, if you want to make a pizza box top with a square base out of some cardboard, we need to do some fancy folding.
Here's what we're gonna do: We'll start with a rectangular sheet of cardboard and cut out six 1-inch squares. Then, we'll fold up the sides to create our pizza box top. Now, since we want the base to be a square, we need to find out what size of a rectangle we need.
Let's say the original length of the rectangle is "L" and the original width is "W". When we cut out those 1-inch squares from each corner, we'll have a length of "L - 2" and a width of "W - 2". Are you following along?
Now, when we fold up the sides, the new height becomes 1 inch. So, the volume of our pizza box top is (L - 2)(W - 2)(1). But we also know that the area of the base is 144 square inches, which means (L - 2)(W - 2) = 144.
Now, let's put our thinking caps on and solve this equation. We need to find the values for L and W that satisfy the equation (L - 2)(W - 2) = 144. Are you ready for the grand reveal?
Drumroll, please... The dimensions of the rectangular sheet of cardboard you should use are length = 14 inches and width = 10 inches. Ta-da! That's the cardboard that will give you a pizza box top with a square base and an area of 144 square inches.
So, grab that cardboard, cut out those squares, fold 'em up, and get ready to enjoy some delicious pizza!
To find the dimensions of the piece of cardboard needed to make the pizza box top with a square base, we can follow these steps:
Step 1: Let's assume the length of the rectangular sheet of cardboard is "L" inches, and the width is "W" inches.
Step 2: We are given that six 1-inch squares are cut from the corners and middle sections. This means that the length of the base of the pizza box will be reduced by 2 inches (1 inch from each side), and the width of the base will also be reduced by 2 inches.
So the length of the base of the pizza box will be L - 2 inches, and the width will be W - 2 inches.
Step 3: The area of the base of the pizza box is given as 144 square inches. Since the base is square-shaped, we can set up the equation:
(L - 2) * (W - 2) = 144
Step 4: Simplify the equation:
Expansion: L * W - 2L - 2W + 4 = 144
Rearrange: L * W - 2L - 2W = 140
Step 5: We need to find the dimensions of the rectangular piece of cardboard, so let's factorize the equation by grouping:
L * W - 2L - 2W - 140 = 0
(L * W - 2L) - (2W + 140) = 0
L * (W - 2) - 2 * (W + 70) = 0
L * (W - 2) - 2 * (W - (-70)) = 0
Step 6: From the grouped equation, we can conclude that either L = 0, or W - 2 = 0, or W - (-70) = 0.
Since the length or width cannot be zero in this context, we can ignore L = 0.
Step 7: Solve for W from the equation W - 2 = 0:
W - 2 = 0
W = 2
Step 8: Substitute the value of W = 2 into the grouped equation to solve for L:
L * (2 - 2) - 2 * (2 - (-70)) = 0
0 - 2 * 72 = 0
-144 = 0 (Not a valid solution)
Step 9: As we can see, L = 0 is not a valid solution. So there is no valid solution for W = 2.
Step 10: Therefore, there is no dimension of the rectangular cardboard sheet that can be used to make the pizza box top with a square base of area 144 square inches.
To solve this problem, we need to determine the dimensions of the rectangular sheet of cardboard that will result in a pizza box with a square base.
Let's start by assuming the length of the rectangular sheet is L inches and the width is W inches.
The pizza box is formed by cutting six 1-inch squares from the corners and the middle sections. This means that the length and width of the base will be reduced by 2 inches each (1 inch from each end) due to these cuts.
Therefore, the length of the base of the pizza box will be (L - 2) inches, and the width will be (W - 2) inches.
The area of the base of the pizza box is given as 144 in^2. So, we can write the equation:
(L - 2) * (W - 2) = 144
Now, let's solve this equation for L and W to find the dimensions of the rectangular sheet of cardboard.
Expand the equation:
LW - 2L - 2W + 4 = 144
Rearrange the terms:
LW - 2L - 2W - 140 = 0
Factor the equation by grouping:
(LW - 2L) - (2W + 140) = 0
Factor out L from the first group and -2 from the second group:
L(W - 2) - 2(W + 70) = 0
Now, we can see that we have a common factor of (W - 2):
(W - 2)(L - 2) - 2(W + 70) = 0
At this point, we can see that one possible solution is W - 2 = 0, which gives us W = 2. However, a pizza box with a width of only 2 inches would not be practical.
So, let's consider the other possible solution, which is:
(L - 2) - 2(W + 70) = 0
L - 2 - 2W - 140 = 0
L - 2W = 142
Now, we have an equation with L - 2W = 142, which we can use. We need to find a pair of values (L, W) that satisfies this equation and also gives us a base area of 144 in^2.
One possible solution is L = 18 and W = 8, because 18 - 2(8) = 18 - 16 = 2, and (18 - 2)(8 - 2) = 16 * 6 = 96, which is smaller than the given area of 144.
Therefore, a rectangular sheet of cardboard measuring 18 inches by 8 inches should be used to make the pizza box.
the base must be x-2 by x-2
(x-2)^2 = 144
x-2 = ±√144 = ±12 , but obviously x must be positive
x-2=12
so the base should be 14 by 14
make a sketch to verify.