Problem 76 - Elementary Algebra

Question

For each of the following problems, use \(3.14\) as an approximation for \(\pi\). Your calculator should be of some help with these problems. Find the area, to the nearest square inch, of each of these pizzas: 10-inch diameter, 12-inch diameter, 14 -inch diameter.

Step-by-Step Solution

Answer

79 sq. in., 113 sq. in., 154 sq. in.

Step 1: Understanding the formula for area of a circle

To find the area of a circle, we use the formula: \( A = \pi r^2 \), where \( r \) is the radius of the circle. For each pizza, we will use this formula to calculate the area.

Step 2: Calculating the radius of each pizza

The radius is half the diameter. For a 10-inch pizza, the radius \( r = \frac{10}{2} = 5 \) inches. For a 12-inch pizza, \( r = \frac{12}{2} = 6 \) inches. For a 14-inch pizza, \( r = \frac{14}{2} = 7 \) inches.

Step 3: Calculating area for 10-inch diameter pizza

Using the formula \( A = \pi r^2 \) with \( r = 5 \), the area is \( A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \). Rounding to the nearest square inch, the area is 79 square inches.

Step 4: Calculating area for 12-inch diameter pizza

With \( r = 6 \), the area is \( A = 3.14 \times 6^2 = 3.14 \times 36 = 113.04 \). Rounding to the nearest square inch, the area is 113 square inches.

Step 5: Calculating area for 14-inch diameter pizza

With \( r = 7 \), the area is \( A = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \). Rounding to the nearest square inch, the area is 154 square inches.

Key Concepts

Geometry

Geometry is an area of mathematics that studies sizes, shapes, and the properties of space. Understanding geometry helps us visualize the world around us, especially when it comes to practical applications like calculating areas and volumes.

To measure the area of a circle, which is a fundamental concept in geometry, we must first understand the terminology.

• Radius: This is the distance from the center of the circle to any point on its edge. Half of the diameter.
• Diameter: The total length from one side of the circle to the other, passing through the center.
• Circumference: The total distance around the circle, although not directly used in area calculations for circles.

These concepts together form the basis for understanding how to calculate different characteristics of circles, such as the area.

Circle Area Calculation

To calculate the area of a circle, we use the formula: \[A = \pi r^2\]Here, \(A\) represents the area, \(\pi\) is a mathematical constant approximately equal to 3.14159 (commonly approximated as 3.14), and \(r\) is the radius of the circle.

This formula arises because the area of a circle is related to its size and the value of \(\pi\). With this formula:

• First, identify the radius by halving the diameter (e.g., a 10-inch diameter pizza has a 5-inch radius).
• Then, square the radius (multiply it by itself).
• Finally, multiply the squared value by \(\pi\), in this case, using the approximation of 3.14.

By following these steps, you can find the area for any circle, whether it's a whole pizza or any circular object in your environment.

Pi Approximation

Pi, denoted as \(\pi\), is a mathematical constant that describes the ratio of a circle's circumference to its diameter. It is an irrational number and cannot be expressed exactly as a simple fraction.

In many practical applications, \(\pi\) is approximated to make calculations more accessible. The most common approximation used is 3.14. However, more precise values may be used such as 22/7 or even more decimals, depending on the level of accuracy required.

• Approximation: Choosing 3.14 as we do in many school exercises is sufficient for basic calculations.
• Applications: In advanced mathematics, physics, engineering, and computer graphics, more precise approximations of \(\pi\) might be needed.

Remember, approximating \(\pi\) helps simplify calculations while still retaining a reasonable level of accuracy for everyday use.