Problem 62 - College Physics

Question

Tony's Pizza Palace sells a medium 9.0 -in. (diameter) pizza for \(\$ 7.95,\) and a large 12 -in. pizza for \(\$ 13.50 .\) Which pizza is the better buy?

Step-by-Step Solution

Answer

The large pizza is the better buy, as it has a lower cost per square inch.

Step 1: Calculate the area of the medium pizza

The area of a circle is calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle. For the medium pizza, the diameter is 9 inches, so the radius is \( \frac{9}{2} = 4.5 \) inches. The area is \( \pi \times (4.5)^2 = 20.25\pi \) square inches.

Step 2: Calculate the cost per square inch for the medium pizza

To find the cost per square inch, divide the price of the medium pizza by its area. The cost is \( \frac{7.95}{20.25\pi} \). Calculating this gives a cost per square inch of approximately \( 0.1255 \) dollars.

Step 3: Calculate the area of the large pizza

For the large pizza, the diameter is 12 inches, so the radius is \( \frac{12}{2} = 6 \) inches. The area is \( \pi \times 6^2 = 36\pi \) square inches.

Step 4: Calculate the cost per square inch for the large pizza

Divide the price of the large pizza by its area to find the cost per square inch. The cost is \( \frac{13.50}{36\pi} \). Calculating this gives a cost per square inch of approximately \( 0.1197 \) dollars.

Step 5: Compare the cost per square inch of both pizzas

Compare the cost per square inch calculated for both pizzas. The medium pizza costs approximately \( 0.1255 \) dollars per square inch, and the large pizza costs approximately \( 0.1197 \) dollars per square inch.

Key Concepts

Area of a Circle

Understanding how to calculate the area of a circle is a fundamental math skill that is useful in many real-life situations. Here, we are dealing with two circular pizzas with different sizes. The key formula for the area of a circle is given by \[ A = \pi r^2 \]where \( A \) represents the area, and \( r \) is the radius of the circle. The radius is always half the diameter of the circle.
For example, in Tony's Pizza Palace, we have a medium pizza with a 9-inch diameter. To find the radius, divide the diameter by 2:
- Medium pizza: \( r = \frac{9}{2} = 4.5 \) inchesThen, substitute the radius back into the formula to find the area:- Medium pizza area: \( A_{medium} = \pi \times (4.5)^2 = 20.25\pi \) square inches
Similarly, for the large pizza with a 12-inch diameter:- Large pizza: \( r = \frac{12}{2} = 6 \) inches
- Large pizza area: \( A_{large} = \pi \times 6^2 = 36\pi \) square inches. Calculating the area this way helps in making exact comparisons between the sizes of the two pizzas.

Cost Efficiency Calculation

Cost efficiency is about getting the most value for the money you spend. In this context, we are determining which pizza gives more pizza area for the dollar spent.
To do this, we calculate the cost per square inch of pizza. This value tells us how much each square inch of pizza costs, providing a basis for comparison between the medium and large pizza.
To find this, divide the price of each pizza by its calculated area:- Medium pizza: - Price: \( \\(7.95 \) - Area: \( 20.25\pi \) square inches - Cost per square inch: \( \frac{7.95}{20.25\pi} \approx 0.1255 \) dollars- Large pizza: - Price: \( \\)13.50 \) - Area: \( 36\pi \) square inches - Cost per square inch: \( \frac{13.50}{36\pi} \approx 0.1197 \) dollars
By doing these calculations, we can see that the large pizza offers a lower cost per square inch, making it a better option for cost efficiency. This demonstrates practical use of mathematics in everyday decisions, like choosing the most economical food purchase.

Pizza Size Comparison

When evaluating two different sizes of pizzas, it's essential to compare them based not only on their price but on the area they cover and the cost efficiency. Let's look at the elements to consider when making such pizza size comparisons.
A larger pizza doesn’t always cost twice as much as a smaller one, and it doesn't necessarily provide twice the area either. However, understanding the relationship between pizza size and price can help make informed decisions.
In this scenario:- Medium pizza: Offers a total area of \( 20.25\pi \) square inches at a cost of \( \\(7.95 \), resulting in a cost of about \( 0.1255 \) dollars per square inch.- Large pizza: Offers a significantly larger total area of \( 36\pi \) square inches at \( \\)13.50 \). Despite the higher price, the cost per square inch is less at approximately \( 0.1197 \) dollars.
By understanding pizza dimensions and doing simple calculations, buyers can recognize that the large pizza, in this case, provides better value for money. This is a great example of smart consumer choices through mathematical reasoning.