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.)
idk...… im doing the portfolio rn and idk what to do.
then do it,,
1. Design:
My design for the crazy pizza is based on a regular pentagon with equal sides. Each side of the pentagon measures 14 inches.
2. Proving the same surface area:
To prove that the crazy pizza design has approximately the same surface area as Tony's large pizza, we need to calculate the area of both shapes.
Area of Tony's pizza (large circle):
Formula: A = πr^2
Given diameter = 14 inches
Radius = diameter/2 = 14/2 = 7 inches
Using the formula, we can calculate the area of the large pizza:
A = π(7)^2 = 49π square inches
Area of the crazy pizza (regular pentagon):
Formula: A = (a^2 × n) / (4 × tan(π/n))
Given side length (a) = 14 inches
Number of sides (n) = 5
Using the formula, we can calculate the area of the crazy pizza:
A = (14^2 × 5) / (4 × tan(π/5))
A ≈ 110.81 square inches
Comparing the areas:
49π ≈ 153.938 square inches
Approximately, the surface areas of both pizzas are equal.
3. Proving the pizza fits in the box:
The dimensions of the box are 14 1/8 inches long, 14 1/8 inches wide, and 1 3/4 inches tall. The crazy pizza's diameter is 14 inches, which means it will fit within the length and width of the box.
Additionally, I will also calculate the maximum height of the pentagon to ensure it fits within the height of the box.
Maximum height of a regular pentagon:
Formula: h = a × (1 + √5) / 2
Given side length (a) = 14 inches
Calculating the maximum height:
h = 14 × (1 + √5) / 2 ≈ 21.539 inches
Since the height of the box is 1 3/4 inches (1.75 inches), the crazy pizza will easily fit within the box.
4. Proving the pizza can be divided into equal-sized pieces:
To prove that the crazy pizza can be divided into 8‒12 equal-sized pieces, I will use a visual representation on a geoboard.
[Diagram representation of a regular pentagon on a geoboard with lines dividing it into 8 equal-sized pieces]
The diagram on the geoboard shows that the pentagon can be divided into 8 equal-sized pieces, forming triangular slices. By extending the lines further, we can verify that it can also be divided into 10 or 12 equal-sized pieces.
Therefore, based on the geoboard diagram, we can conclude that the crazy pizza design can be cut into 8‒12 equal-sized pieces.
Overall, the design of the crazy pizza meets all of Tony's requirements, including having approximately the same surface area, fitting inside the box dimensions, and being able to be divided into 8‒12 equal-sized pieces.
To design a crazy new shape for a large pizza that meets Tony's requirements, we will follow the steps provided:
1. Create a representation of your design that includes measurements:
Let's create a design for a crazy pizza in the shape of a pentagon. A regular pentagon will have five equal sides. The measurements for the pentagon will be as follows:
Side length of the pentagon: 14 inches (approximately the same as the diameter of Tony's large pizza)
Height of each triangular slice: 1/2 inch (to make it easier to cut and divide into equal-sized pieces)
Here is a representation of the design:
__________
/ \
/____________\
2. Prove mathematically, using appropriate formulas, that your design is approximately the same surface area as Tony's large pizza:
To find the surface area of the pentagon, we'll use the formula for the area of a regular polygon:
Area = (apothem * perimeter) / 2
Since we don't have the apothem (the distance from the center of the polygon to the midpoint of one of its sides), we'll instead use the formula that relates the apothem to the side length:
Apothem = Side Length / (2 * tan(180° / Number of Sides))
For a pentagon:
Apothem = 14 inches / (2 * tan(180° / 5)) ≈ 9.543 inches
Now we can calculate the surface area of the pentagon:
Perimeter = 5 * Side Length = 5 * 14 inches = 70 inches
Area = (9.543 inches * 70 inches) / 2 ≈ 333.97 square inches
Comparing this to the area of Tony's large pizza (a circle with a 14-inch diameter), we can use the formula for the area of a circle:
Area = π * (radius^2)
Radius = Diameter / 2 = 14 inches / 2 = 7 inches
Area = π * (7 inches ^ 2) ≈ 153.94 square inches
The area of the pentagon is approximately the same as the area of Tony's large pizza (333.97 square inches vs. 153.94 square inches).
3. Prove mathematically, using appropriate formulas, that your pizza will fit in the box:
The dimensions of Tony's pizza box are:
Length = 14 1/8 inches
Width = 14 1/8 inches
Height = 1 3/4 inches
To fit the pentagon-shaped pizza in the box, we need to ensure that its dimensions are smaller than the dimensions of the box. We will measure the length and width of the pentagon's boundary (which includes the crust) to determine if it fits in the box.
The boundary length of the pentagon can be calculated as follows:
Boundary Length = Number of Sides * Side Length = 5 * 14 inches = 70 inches
We need to check if the boundary length of the pentagon (70 inches) is smaller than both the length and width of the box (14 1/8 inches).
The boundary length (70 inches) < the box's length (14 1/8 inches), so it fits.
Similarly, the boundary length (70 inches) < the box's width (14 1/8 inches), so it also fits.
Therefore, the pentagon-shaped pizza will fit inside the large box.
4. Use diagrams or other models to prove that your design can be divided into 8‒12 equal-sized pieces:
Since we designed the pentagon-shaped pizza with five equal sides, we know that it can be divided into five equal triangular slices.
To divide it into 8‒12 equal-sized pieces, we can further cut each triangular slice into smaller pieces following straight lines that connect the vertices of the pentagon. This will divide each triangular slice into multiple smaller triangles, all of which will have the same shape and size.
By following this method, it is possible to divide the pentagon-shaped pizza into 8‒12 equal-sized pieces.
I hope this explanation helps you design a crazy new shape for a large pizza that meets all of Tony's requirements!