Problem 60 - Introductory Algebra for College Students

Question

Use the formulas for the area and the circumference of a circle in Table 2.4 on page 170 to solve. Unless otherwise indicated, round all circumference and area calculations to the nearest whole number. Which one of the following is a better buy: a large pizza with a 16 -inch diameter for \(\$ 12.00\) or two small pizzas, each with a 10 -inch diameter, for \(\$ 12.00 ?\)

Step-by-Step Solution

Answer

The large pizza is a better buy as it provides more area for the same price.

Step 1: Obtain Radius of Each Pizza

The diameters of the pizzas are given, but the formula for area requires the radius. So the first step is to find the radius of each pizza. This can be done by simply halving the diameter. Hence, the radius of the large pizza is \(16/2 = 8\) inches and the radius of the small pizza is \(10/2 = 5\) inches.

Step 2: Calculate the Area of Each Pizza

The next step is to calculate the area of each pizza using the formula \(\pi r^{2}\). For the large pizza, this gives \(\pi (8)^{2} =64\pi\) square inches. For each small pizza, it gives \(\pi (5)^{2} =25\pi\) square inches.

Step 3: Compare the Total Area to the Price

Finally, compare the total area offered by each option to the price. Keep in mind that for the two small pizzas, you have to add the areas of the two. So the total area for the two small pizzas is \(2*25\pi =50\pi\) square inches. Since both options cost $12, the large pizza—which has an area of 64π square inches—is a better buy than two small pizzas.

Key Concepts

Area of a Circle

The area of a circle tells us how much space is enclosed within its boundary. It's a measure of the two-dimensional space within the circle's perimeter. To calculate the area, we use the formula:\[ A = \pi r^{2} \]where \(A\) stands for area and \(r\) represents the radius of the circle. The symbol \(\pi\) (Pi) approximately equals 3.14159, but is often simplified to 3.14 for easier calculations.
In the context of a pizza, calculating the area lets you determine how much pizza you actually get to eat. Larger areas mean more pizza. For example, a pizza with an 8-inch radius will have an area of:\[ A = \pi (8)^{2} = 64\pi \] square inches.

• This means there are 64π square inches of delicious pizza to enjoy.

Circumference of a Circle

The circumference of a circle is the total distance around its outer edge. Think of it as the perimeter of the circle. This is what you'd measure if you wrapped a string around a circle's edge. To determine the circumference, we utilize the formula:\[ C = 2\pi r \]where \(C\) is the circumference and \(r\) is the radius. Alternatively, you could use the formula based on the diameter:\[ C = \pi d \]Here, \(d\) is the circle's diameter.
Knowing the circumference can also tell you how much crust you might encounter when enjoying a pizza, or how far you'll need to walk if you trek around a circular pond. When handling pizza, remember, even though circumference is interesting, the area is what matters when you want to know how much you get per bite.

Radius

The radius is a crucial part of a circle's geometry and plays a vital role in calculating both the area and circumference. It represents the distance from any point on the circle to its center. The radius is half of the diameter:\[ r = \frac{d}{2} \]Understanding this can make it easier to handle circle-related problems, especially when you start with the diameter.

• For a pizza with a 16-inch diameter, the radius comes out to be 8 inches.
• Finding the radius is often the first step in solving any circle problem, as it unlocks all other calculations.

Diameter

The diameter of a circle is the longest distance across the circle, passing through the center. It is essentially twice the length of the radius:\[ d = 2r \]Knowing the diameter is useful, especially when you need to quickly find out the radius or if you directly want to calculate the circumference using \(\pi d\). For our pizza example, pizzas are often listed by their diameter. This is common in everyday scenarios, like ordering pizza, where you might find 10-inch or 16-inch pizzas.

• The diameter tells you how big the pizza will appear when placed in front of you.
• It also helps determine the cooking time, since larger pizzas might need more time in the oven.