Problem 23 - College Algebra with Modeling and Visualization

Question

Find the circumference and area of the circle. Approximate each value to the nearest tenth when appropriate. \(r=19\) inches

Step-by-Step Solution

Answer

The circumference is approximately 119.4 inches, and the area is approximately 1134.1 square inches.

Step 1: Identify Circle Dimensions

First, understand that the circle has a given radius of \( r = 19 \) inches. The radius is the distance from the center of the circle to any point on its edge.

Step 2: Formula for Circumference

To find the circumference \( C \) of a circle, use the formula \( C = 2\pi r \). This formula uses \( \pi \approx 3.14159 \), and \( r \) is the radius of the circle.

Step 3: Calculate the Circumference

Substitute the given radius into the circumference formula: \[ C = 2 \times \pi \times 19 \approx 2 \times 3.14159 \times 19. \] Calculate to get: \( C \approx 119.38052 \). Thus, the circumference is approximately 119.4 inches when rounded to the nearest tenth.

Step 4: Formula for Area

To find the area \( A \) of a circle, use the formula \( A = \pi r^2 \). This requires squaring the radius and multiplying by \( \pi \).

Step 5: Calculate the Area

Substitute the given radius into the area formula: \[ A = \pi \times 19^2 = \pi \times 361. \] Calculate to get: \( A \approx 3.14159 \times 361 \approx 1134.11479 \). Thus, the area is approximately 1134.1 square inches when rounded to the nearest tenth.

Key Concepts

Circumference of a Circle

The circumference of a circle is the total distance around the circle. Imagine if you walked around the edge of a giant, perfectly round garden. That journey is similar to finding the circumference. To find this measurement, we use the formula:

• \( C = 2\pi r \)

Here, \( C \) stands for circumference, \( r \) is the radius (or the distance from the center of the circle to its edge), and \( \pi \) is a mathematical constant approximately equal to 3.14159.To calculate the circumference, simply plug in the radius. In our exercise, the radius is 19 inches.
The calculation process looks like this:

• Replace \( r \) in the formula with 19: \( C = 2 \times \pi \times 19 \).
• Using \( \pi \approx 3.14159 \), the calculation becomes: \( C \approx 2 \times 3.14159 \times 19 \).
• This simplifies to \( C \approx 119.38052 \).
• When rounded to the nearest tenth, the circumference is 119.4 inches.

Now, after rounding, you have the complete measure of the circle's edge in inches.

Area of a Circle

The area of a circle gives us the size of the surface enclosed by the circle's edge. Think of it as how much space is inside a round pizza. The formula to find this space is:

• \( A = \pi r^2 \)

In this formula, \( A \) stands for area, \( r \) for radius, and \( \pi \) is again approximately 3.14159.
To find the area, you multiply \( \pi \) by the square of the radius. In our problem, the radius provided is 19 inches, so our steps go like this:

• First, square the radius: \( 19^2 = 361 \).
• Next, multiply this result by \( \pi \): \( A = \pi \times 361 \).
• This results in \( A \approx 3.14159 \times 361 \approx 1134.11479 \).
• After rounding to the nearest tenth, you find the area is approximately 1134.1 square inches.

By following these steps, you can determine the amount of space within any circle given its radius.

Radius of a Circle

The radius of a circle is a crucial part of its anatomy. It's the distance from the center of the circle to any point along its edge, a straight line that is half the length of the circle's diameter. Imagine cutting a giant cookie in half. The straight line from the center to the edge is the radius.Knowing the radius is essential in every calculation involving a circle since both circumference and area use the radius in their formulas.

• The circumference uses the formula \( C = 2\pi r \). Notice the \( r \) being part of the equation.
• The area utilizes \( A = \pi r^2 \), and again, the \( r \) shows up prominently.

In our previous exercise, the radius given was 19 inches, which directly influenced the calculations for both circumference and area. Converting these measurements into the final solutions shows the power of understanding and accurately using the radius to unlock circle dimensions.