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Chapter 1: Problem 53

A pizzeria offers a 9 -inch-diameter pizza for \(\$ 12\) and an 18-inch-diameter pizza for \(\$ 24\). Are both offerings equally economical? If not, which is the better deal? Explain your reasoning.

Short Answer

Expert verified
The 18-inch pizza is the better deal.

Step by step solution

01

Calculate the radius of both pizzas

The radius is half of the diameter. For the 9-inch pizza, the radius is \ \( \frac{9}{2} = 4.5 \ \) inches. For the 18-inch pizza, the radius is \ \( \frac{18}{2} = 9 \ \) inches.
02

Calculate the area of both pizzas

The area of a circle is given by the formula \ \( A = \pi r^2 \). For the 9-inch pizza: \ \( A = \pi (4.5)^2 = 20.25\pi \). For the 18-inch pizza: \ \( A = \pi (9)^2 = 81\pi \).
03

Calculate the cost per square inch

For the 9-inch pizza: \ \( \frac{12}{20.25\pi} = \frac{12}{63.62} \ = 0.19\ \text{dollars per square inch} \). For the 18-inch pizza: \ \( \frac{24}{81\pi} \ = \frac{24}{254.47} \ = 0.09 \text{dollars per square inch} \).
04

Compare the cost effectiveness

The 18-inch pizza costs \ \( 0.09 \text{dollars per square inch} \), while the 9-inch pizza costs \ \( 0.19 \text{dollars per square inch} \). Therefore, the 18-inch pizza is the better deal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diameter and Radius
The diameter and radius are crucial in understanding the size of a pizza. The diameter is the distance across the pizza through its center. The radius, however, is half of the diameter, representing the distance from the center to the edge.
For example, if a 9-inch-diameter pizza is offered, its radius would be half of 9 inches, which is 4.5 inches. Similarly, for an 18-inch-diameter pizza, the radius is half of 18 inches, which is 9 inches.
Understanding these concepts helps us move on to more complex calculations involving area, as the radius is a key component in finding the area of a circular object like a pizza.
Area Calculation
After finding the radius, the next step is calculating the area. This is done using the formula for the area of a circle: \( A = \pi r^2 \).
Here, \( A \) is the area, \( \pi \) is approximately 3.14159, and \( r \) is the radius. So, for a 9-inch pizza with a radius of 4.5 inches, the area calculation becomes: \( A = \pi (4.5)^2 = 20.25\pi \). For an 18-inch pizza with a radius of 9 inches, the area is: \( A = \pi (9)^2 = 81\pi \).
These area values help us determine how much pizza you get, but alone, they don't yet tell us about the cost efficiency.
Cost Efficiency
Cost efficiency helps us determine which pizza gives more value for money. To find this, we calculate the cost per square inch.
For the 9-inch pizza priced at \$12, the cost per square inch is: \(\frac{12}{20.25\pi} = \frac{12}{63.62} = 0.19 \) dollars per square inch. For the 18-inch pizza priced at \$24, the cost per square inch is: \(\frac{24}{81\pi} = \frac{24}{254.47} = 0.09 \) dollars per square inch.
By comparing these values, it’s clear that the 18-inch pizza, with a cost of 0.09 dollars per square inch, is more economical than the 9-inch pizza, which costs 0.19 dollars per square inch. Thus, for a better deal, go for the larger pizza!

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