Problem 63 - College Physics

Question

Two students go into Tony's Pizza Palace and order a 12-in. (diameter) pizza. Shortly thereafter, the waitress brings an 8 -in. pizza special. She explains that the 12 -in. pizza was given to someone else by mistake and they could have the 8 -in. now and she would bring another 8 in. shortly to make up for the missing 12 -in. pizza. Was this a good deal?

Step-by-Step Solution

Answer

The deal is not good; they receive less pizza with the two 8-in. pizzas.

Step 1: Calculate the Area of the 12-in. Pizza

The formula for the area of a circle is \( A = \pi r^2 \), where \( r \) is the radius. The 12-in. pizza has a diameter of 12 inches, so its radius \( r = \frac{12}{2} = 6 \) inches. Thus, the area of the 12-in. pizza is \[ A_{12} = \pi (6^2) = 36\pi \text{ square inches}.\]

Step 2: Calculate the Area of an 8-in. Pizza

The diameter of each 8-in. pizza is 8 inches, so the radius \( r = \frac{8}{2} = 4 \) inches. Using the formula for the area of a circle, the area of one 8-in. pizza is \[ A_8 = \pi (4^2) = 16\pi \text{ square inches}.\]

Step 3: Calculate the Total Area of the Two 8-in. Pizzas

Since the waitress is offering two 8-in. pizzas, we need to calculate the total area. This is \[ 2 \times 16\pi = 32\pi \text{ square inches}.\]

Step 4: Compare the Areas

The area of the 12-in. pizza is \( 36\pi \), while the total area of the two 8-in. pizzas is \( 32\pi \). Thus, the 12-in. pizza has more area than the combined two 8-in. pizzas.

Key Concepts

Understanding Geometry in Circle Area Calculation

Understanding the geometry of a circle is essential when dealing with circle area calculations. In geometry, a circle is a simple, closed shape. Every point on the circle is equidistant from its center, which is the circle's defining feature. Calculating the area of a circle is one of the basic principles in geometry. This formula, \( A = \pi r^2 \), requires knowing the circle's radius. An important part of understanding circle geometry is grasping that changes in the radius or diameter will indeed change the circle's area dramatically due to the square in the formula. The larger the radius, the greater the area of the circle, showcasing the unique properties of circular geometry.

The Relationship Between Radius and Diameter

The concepts of radius and diameter are fundamental in understanding a circle's properties and how to calculate its area. The radius of a circle is the distance from its center to any point on its boundary. The diameter is twice the radius, extending from one edge of the circle, passing through the center, to the opposite edge.Here are some simple points:

• Radius = Diameter / 2
• Diameter = 2 * Radius

In our exercise, for example, the 12-inch pizza has a diameter of 12 inches. Hence, the radius is 6 inches (\( \frac{12}{2} \)). This precise relationship is crucial, as it directly impacts the formula used to find the circle's area.

Area Comparison: Making Sense of Size

When it comes to assessing size, comparing areas gives a clear picture of which object is larger. In our pizza problem, comparing a 12-inch pizza with two 8-inch pizzas requires understanding how to calculate and then evaluate these areas.The area of a 12-inch pizza is \( 36\pi \) square inches (since \( A = \pi r^2 \) with \( r = 6 \)). For one 8-inch pizza, the area calculates to \( 16\pi \) square inches (with \( r = 4 \)). Thus, two 8-inch pizzas provide \( 32\pi \) square inches (\( 2 \times 16\pi \)). Clearly, the 12-inch pizza has a larger area compared to the two smaller ones, allowing us to conclude that the original 12-inch pizza deal offers more value in terms of area.

Applying Mathematical Problem Solving

Mathematical problem solving involves a structured approach to finding solutions, which is clearly demonstrated in this exercise. To effectively tackle such problems, one should:

• Identify the known quantities: diameters of the pizzas here.
• Compute the required values: areas of the pizzas using \( A = \pi r^2 \).
• Compare the results critically: here, compare the two differing pizza scenarios.

By systematically calculating and comparing the areas, one seamlessly arrives at the solution, which in our case illustrates that a single larger pizza usually isn't equaled by simply two smaller ones in size. This structured, logical approach to problem solving reinforces comprehension and accuracy in mathematical tasks.