Problem 51 - Algebra A Combined Function

Question

Maria's Pizza sells one 16 -inch cheese pizza or two 10 -inch cheese pizzas for \(\$ 9.99 .\) Determine which size gives more pizza. \(\left(A=\pi r^{2}\right)\)

Step-by-Step Solution

Answer

The 16-inch pizza gives more pizza.

Step 1: Identify the radii

To calculate the area of a circle, we need the radius. For a 16-inch pizza, the diameter is 16 inches, so the radius is 8 inches. For each 10-inch pizza, the diameter is 10 inches, so the radius is 5 inches.

Step 2: Calculate the area of the 16-inch pizza

The formula for the area of a circle is given by \(A = \pi r^2\). Substituting the radius of the 16-inch pizza, we get \(A_{16} = \pi \times 8^2 = 64\pi\).

Step 3: Calculate the area of a single 10-inch pizza

Using the same formula, the area of a 10-inch pizza is \(A_{10} = \pi \times 5^2 = 25\pi\).

Step 4: Calculate the total area for two 10-inch pizzas

Since there are two 10-inch pizzas, the total area is \(2 \times 25\pi = 50\pi\).

Step 5: Compare the areas

The area of the 16-inch pizza is \(64\pi\) square inches, while the combined area of the two 10-inch pizzas is \(50\pi\) square inches. Since \(64\pi > 50\pi\), the 16-inch pizza provides more pizza.

Key Concepts

Radius of a Circle

Understanding the concept of the radius is essential in circle measurements. The radius of a circle is the distance from the center of the circle to any point on its edge. This distance is crucial in many calculations involving circles.
For example, if you have a circle with a diameter of 16 inches, you can find the radius by dividing the diameter by two. Thus, for a circle with a diameter of 16 inches, the radius is 8 inches.
Similarly, for a circle with a diameter of 10 inches, the radius is 5 inches. Remember, the diameter is simply twice the radius.

Circle Area Formula

The area of a circle can be found using the formula:

• \( A = \pi r^2 \)

In this formula, \( A \) stands for the area, \( \pi \) is a mathematical constant approximately equal to 3.14159, and \( r \) is the radius of the circle.

To find the area, you square the radius and multiply by \( \pi \). For instance, if the radius of a circle is 8 inches, the area will be

• \( A = \pi \times 8^2 = 64\pi \) square inches

Using this formula, you'll easily calculate areas for any circle!

Comparison of Areas

When comparing areas of different circles, it's important to use consistent units and measurements. In the context of our pizza problem, we're comparing the area of a larger pizza to the combined area of two smaller ones.

Using the circle area formula, we found that the single 16-inch pizza has an area of \( 64\pi \) square inches. Meanwhile, one 10-inch pizza has an area of \( 25\pi \) square inches. Since there are two 10-inch pizzas, their total area is \( 50\pi \) square inches.

• This means that \( 64\pi > 50\pi \),
• in turn showing the larger pizza provides more food.

Pizza Measurement

Measuring pizzas, or any circle-shaped food, involves understanding both the size and the area. Consumers often encounter choices: buy one larger item or several smaller items?

Opting for a larger pizza might seem more expensive at first, but as seen from our calculations, the bigger pizza provides more area, thus more pizza.

• This stems from the exponential relationship between the radius and area in circle geometry.
• For maximizing value, it’s effective to calculate and compare areas.

Next time you're faced with a similar choice, remember to calculate and find which option offers the most for your money!