I only need help with A and B
A pizzeria is putting on a ‘pie-by-length’ event where customers
can order a rectangular pizza that is 6″ wide by the length of
their choice. Customers can also order circular pizzas of several
diameters.
a. Solving the equation 6L=(pi)(1/2)^2 will give the length/diameter where the two shapes have the same area. Explain how this
formula was determined.
b. For what length/diameter will the two shapes of pizza have the same area? Round the answer to the nearest tenth of an inch.
c. The pizzeria owner says that when customers order pie-by-length, they feel like they are getting more for less. How might he justify this claim?
6L = π(1/2)^2
L = π/24
so the long pizza has area π/24 * 6 = π/4
The round pizza has area π * (1/2)^2 = π/4
Those are pretty small pizzas!
a. To determine the formula for finding the length/diameter where the rectangular and circular pizzas have the same area, we need to equate the areas of both shapes.
The area of a rectangle is given by the formula A = length × width. In this case, the width is fixed at 6 inches.
The area of a circle is given by the formula A = πr^2, where r is the radius. Since the diameter is twice the radius, we can rewrite the formula as A = π(d/2)^2, where d is the diameter.
Since we want the two shapes to have the same area, we can set up the equation:
6L = π(d/2)^2
Here, 6L represents the area of the rectangular pizza and π(d/2)^2 represents the area of the circular pizza.
b. To find the length/diameter where the two shapes have the same area, we can solve the equation 6L = π(d/2)^2 for d.
Here's how you can solve it:
1. Multiply both sides of the equation by 2 to eliminate the fraction: 12L = πd^2.
2. Divide both sides by π to isolate d^2: (12L)/π = d^2.
3. Take the square root of both sides to solve for d: d = √((12L)/π).
4. Round the answer to the nearest tenth of an inch.
c. The pizzeria owner's claim that customers feel like they are getting more for less when ordering pie-by-length can be justified by comparing the areas of the two shapes.
Since the rectangular pizza has a fixed width of 6 inches, customers have the freedom to choose any length they want. This means they can order pizzas with different lengths but the same area as the circular pizzas.
In other words, for the same area, the rectangular pizza can have a longer length compared to the diameter of the circular pizza. This gives the perception that customers are getting more pizza (length) for the same area as the circular pizza.
By offering pie-by-length, the pizzeria is catering to customer preferences and providing the illusion of receiving more pizza for less.