Problem 59 - Introductory and Intermediate Algebra for College Students 4th

Question

Which one of the following is a better buy: a large pizza with a 14 -inch diameter for \(\$ 12.00\) or a medium pizza with a 7 -inch diameter for \(\$ 5.00 ?\)

Step-by-Step Solution

Answer

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

Step 1: Determine the radius of each pizza

Since the diameters are given and the radius is simply half the diameter, we find that the large pizza has a radius of 14/2 = 7 inches, and the medium pizza has a radius of 7/2 = 3.5 inches.

Step 2: Calculate the area of each pizza

We use the formula for the area of a circle, \( \pi r^{2} \). For the large pizza, this gives us \( \pi * (7)^{2} = 153.94 \) square inches, and for the medium pizza we get \( \pi * (3.5)^{2} = 38.48 \) square inches.

Step 3: Determine the cost per square inch of each pizza

Divide the cost of each pizza by its area. For the large pizza, this gives us $12.00 / 153.94 =~ $0.078 per square inch. For the medium pizza, we find $5.00 / 38.48 =~ $0.13 per square inch.

Step 4: Compare the cost per square inch of both pizzas

The lower the cost per square inch, the better the buy. In this case, the large pizza costs less per square inch, making it the better buy.

Key Concepts

Area of a Circle

Understanding the area of a circle is essential when dealing with problems that involve circular shapes, such as pizzas. The formula to calculate the area is \(A = \pi r^{2}\), where \(r\) stands for the radius of the circle. The radius is the distance from the center of the circle to any point on its edge. Often, problems will provide the diameter (the distance across the circle through the center). Remember, the radius is half the diameter.

In our pizza problem, the formula helps us determine how much surface we are getting from each pizza. It's important to calculate the area accurately to make fair comparisons. Always follow the formula precisely, and don’t forget to use the same units (typically inches or centimeters). This ensures the areas you calculate truly represent the size of the circle.

Cost Per Square Inch

When trying to decide which pizza offers better value, comparing cost per square inch makes sense. To find this, divide the cost of the pizza by its total area. This calculation shows you how much each square inch of pizza costs, giving you a clearer picture of the value you’re getting.

In our example, the large pizza with an area of 153.94 square inches costs $12.00, leading to a cost of approximately $0.078 per square inch. Contrast this with the medium pizza, which, with its smaller area of 38.48 square inches, costs $0.13 per square inch. Clearly, the large pizza offers a better deal in terms of price for each square inch of tasty goodness you’re buying.

Comparing Values

Effectively comparing values involves checking which option gives you more for less. In mathematical or economic terms, you analyze the benefit received per unit cost.

In this exercise, you compare two pizzas, each with their own cost per square inch. While both pizzas provide a delicious meal, you want the most bang for your buck. By identifying that the large pizza costs less per square inch than the medium pizza, you've determined that the larger one offers better value.

When faced with multiple options, always compute which gives you a lower cost per useful unit. This approach is applicable beyond pizzas, such as in housing (price per square foot) or groceries (price per ounce). It helps consumers make informed decisions.

Problem-Solving Steps

Effective problem-solving involves a clear, logical approach to tackling a challenge or question. Follow these steps:

• Identify what you need to find out. For instance, in this pizza problem, the goal is to discover which pizza provides better value based on cost and size.
• Gather all necessary information, such as costs and measurements given in the problem.
• Perform calculations accurately. For areas of circles, apply the formula \(A = \pi r^{2}\), and for cost per square inch, divide the price by the area.
• Compare results logically to determine the answer. In our scenario, the pizza with the lower cost per square inch is the better buy.

Practicing structured problem-solving in various scenarios sharpens your analytical skills and makes it easier to address more complex challenges efficiently.