A large pizza at Tony's Pizzeria is a circle with a 14-inch diameter. Its box is a rectangular prism that is 14 1/8 inches long, 14 1/8 inches wide, and 1 3/4 inches tall. Your job is to design a crazy new shape for a large pizza. It can be any
irregular polygon shape, but it must have at least five sides.
Tony says he will make and sell your crazy pizza if you can prove the following:
• It is approximately the same area as the large pizza he sells now.
• It fits inside the large box listed above.
• It can be cut into 8‒12 equal-sized pieces.
Create one design for a crazy pizza that will meet all of Tony's requirements.
1. Create a representation of your design that includes measurements.
2. Prove mathematically, using appropriate formulas, that your design is
approximately the same surface area as Tony’s large pizza.
3. Prove mathematically, using appropriate formulas, that your pizza will fit in
the box.
4. Use diagrams or other models to prove that your design can be divided into
8‒12 equal-sized pieces. (Tip: You may use graph paper or the Virtual
Geoboard to show how your pizza can be divided into equal-sized pieces.)
sorry it was kinda 4 years ago
To design a new shape for a large pizza that meets Tony's requirements, we need to consider the following:
1. Create a representation of your design that includes measurements:
For example, let's design a crazy pizza in the shape of a hexagon. Each side of the hexagon will measure approximately 14 inches.
2. Prove mathematically, using appropriate formulas, that your design is approximately the same surface area as Tony's large pizza:
To calculate the surface area of a regular hexagon, we can use the formula A = (3√3 x S^2) / 2, where S is the length of each side.
In this case, S = 14 inches. Plugging this value into the formula gives us:
A = (3√3 x 14^2) / 2
A ≈ 211.94 square inches
The surface area of Tony's large circular pizza can be calculated using the formula A = πr^2, where r is the radius of the circle. The diameter of the circular pizza is given as 14 inches, so the radius (r) is half of that, which is 7 inches.
Plugging this value into the formula gives us:
A = π x 7^2 ≈ 153.94 square inches
Comparing the surface area of our hexagonal pizza (approximately 211.94 square inches) to the circular pizza (approximately 153.94 square inches), we can conclude that they are approximately the same size.
3. Prove mathematically, using appropriate formulas, that your pizza will fit in the box:
The dimensions of the box are given as 14 1/8 inches long, 14 1/8 inches wide, and 1 3/4 inches tall. We need to ensure that our hexagonal pizza can fit within these dimensions.
The length and width of the box are both 14 1/8 inches, which is approximately 14.125 inches. The tallest point of our hexagonal pizza will be the distance from the center to one of the corners, which is the length of the apothem.
Using the formula apothem = S / (2 * tan(π / n)), where S is the side length and n is the number of sides, we can calculate the apothem for our hexagon.
In this case, S = 14 inches and n = 6 (for a hexagon).
apothem = 14 / (2 * tan(π / 6))
apothem ≈ 6.429 inches
Since the height of the box is given as 1 3/4 inches, which is approximately 1.75 inches, we can see that our hexagonal pizza will fit within the box.
4. Use diagrams or other models to prove that your design can be divided into 8‒12 equal-sized pieces:
To demonstrate that our hexagonal pizza can be divided into 8‒12 equal-sized pieces, we can use a visual representation. Here is one possible way to divide the hexagon into 8 equal-sized pieces:
- Draw lines from the center of the hexagon to each vertex, dividing it into 6 equal triangles.
- Then, draw lines parallel to the sides of the hexagon, dividing each triangle into two smaller triangles.
- Along these lines, cut the pizza to create 8 equal-sized slices.
You can replicate this process to divide the hexagon into 12 equal-sized pieces by further dividing each of the smaller triangles into two. The key is to divide each side and angle equally to ensure equal-sized slices.
By following these steps and calculations, you can design a hexagonal pizza that meets all of Tony's requirements.