Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

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 the better buy since it provides a larger area for the same price as the two small pizzas.

Step 1: Find the Area of the Large Pizza

The diameter of the large pizza is 16 inches, so its radius is half of that, which is 8 inches. Using the formula for the area of a circle (\(πr^2\)), the area of the large pizza is \(π * 8^2 = 64π\) square inches.

Step 2: Find the Area of Each Small Pizza and Then the Total Area

The diameter of each small pizza is 10 inches, so its radius is half of that, which is 5 inches. The area of one small pizza is \(π * 5^2 = 25π\) square inches. Since there are two small pizzas, the total area is \(2 * 25π = 50π\) square inches.

Step 3: Compare the Total Area of the Large Pizza and the Total Area of the Small Pizzas

The total area of the large pizza is 64π square inches and that of the small pizzas is 50π square inches. Thus, the large pizza has a greater area.

Step 4: Check the Price

Both pizza options cost $12.00, but the large pizza has a greater area. Therefore, the large pizza is the better buy.

Key Concepts

Circle Area Calculation

Understanding how to calculate the area of a circle is fundamental when comparing quantities like pizza sizes. The area of a circle is found using the formula
\( A = \pi r^2 \),
where
\( A \) represents the area,
\( \pi \) is a constant approximately equal to 3.14159, and
\( r \) stands for the radius of the circle. When you know the diameter, the radius is simply half of this length. In the context of ordering pizzas and getting the most for your money, calculating the area gives you a clear understanding of the size of the pizza you are receiving. For the large pizza with a radius of 8 inches, plug the value into the formula to get
\( 64\pi \) square inches. Comparing this to the combined area of two small pizzas, it becomes apparent which provides more pizza per dollar spent.

Radius and Diameter of Circles

A circle's diameter is the longest straight line that can be drawn inside the circle, passing through its center. The radius of a circle, which is half the diameter, extends from the center to any point on the circle's circumference. These measurements are the building blocks of various geometric calculations, especially area. In our pizza example, the diameter of the large pizza is 16 inches, thus the radius is
\( \frac{16}{2} = 8 \) inches.
Similarly, the small pizza with a 10-inch diameter has a radius of
\( \frac{10}{2} = 5 \) inches. Knowing these values is crucial for accurately computing the area, which in turn allows a buyer to make an informed decision when choosing between different pizza sizes and prices.

Value Comparison in Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is a powerful tool for solving problems and making comparisons, such as deciding which pizza to buy. When presented with two options that cost the same, algebra can help determine the better value by comparing quantities, like the total areas of different pizzas. You create an equation to represent each option and compare the results. In our pizza scenario, after calculating the total area of both options, we set up a comparison:
\( 64\pi > 50\pi \).
Even though the pizzas are the same price, algebra tells us that the large pizza with an area of
\( 64\pi \) square inches offers more food than the total area of the two smaller pizzas combined, which is
\( 50\pi \) square inches. Thus, the larger pizza offers the better value for money, a conclusion reached by applying algebraic concepts for value comparison.