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.)
Please help! I need this to be done today!
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To design a crazy new shape for a large pizza, we need to follow Tony's requirements:
1. Design a shape that is approximately the same area as the current large pizza:
To do this, we can calculate the area of Tony's large pizza using the formula for the area of a circle: A = πr², where r is the radius. Since the diameter of the current pizza is 14 inches, the radius is half of that, which is 7 inches. Therefore, the area of the current pizza is A = π(7)² ≈ 153.94 square inches.
To design a shape with a similar area, we can choose an irregular polygon such as a pentagon, hexagon, or heptagon. Let's consider a hexagon for this example.
2. Prove mathematically that your design has approximately the same surface area as Tony's large pizza:
To calculate the area of a regular hexagon, we can use the formula: A = (3√3/2) × s², where s is the length of a side. Since we want the area to be close to 153.94 square inches, we set up the equation: (3√3/2) × s² ≈ 153.94.
To solve for s, we rearrange the equation:
s² ≈ 153.94 × 2 / (3√3)
s² ≈ 307.88 / (3√3)
s² ≈ 56.213
Taking the square root of both sides, we get:
s ≈ √56.213
s ≈ 7.499
So, we can design a regular hexagon with a side length of approximately 7.499 inches, which will have an area close to the area of Tony's large pizza.
3. Prove mathematically that your pizza will fit in the box:
To check if the hexagon pizza will fit inside the box, we need to calculate its maximum diagonal length. For a regular hexagon, the diagonal can be calculated using the formula: d = s × √3, where s is the length of a side.
For our hexagon design, the side length is approximately 7.499 inches, so the diagonal would be:
d = 7.499 × √3 ≈ 13.0 inches.
Since the box's length and width are both 14 1/8 inches, which is greater than 13.0 inches, we can conclude that the hexagon pizza will fit inside the box.
4. Use diagrams or models to prove that your design can be divided into 8-12 equal-sized pieces:
To show how the hexagon pizza can be divided into equal-sized pieces, you can use graph paper or a virtual geoboard. Start by drawing a regular hexagon with the given side length of 7.499 inches. Then, divide it into 8-12 equal-sized triangular slices by drawing lines from the center of the hexagon to each corner.
Using this method, you can visually demonstrate that the hexagon pizza can be cut into equal-sized pieces within the desired range.
Remember to provide visual representations and detailed calculations to support your design and prove that it meets Tony's requirements.