Problem 22 - College Algebra with Modeling and Visualization

Question

Find the circumference and area of the circle. Approximate each value to the nearest tenth when appropriate. \(r=1.5\) feet

Step-by-Step Solution

Answer

Circumference is 9.4 feet; area is 7.1 square feet.

Step 1: Understand the Problem

We are given a circle with a radius, \( r = 1.5 \) feet, and are asked to find both the circumference and the area of this circle. We will use the formulas for the circumference \( C = 2 \pi r \) and the area \( A = \pi r^2 \).

Step 2: Calculate the Circumference

To calculate the circumference of the circle, use the formula: \( C = 2 \pi r \). Substituting the given value for \( r \), we have:\[ C = 2 \pi \times 1.5 = 3 \pi \].Using the approximation \( \pi \approx 3.14159 \), we can compute \( C \approx 3 \times 3.14159 = 9.42477 \). Rounding this to the nearest tenth:\[ C \approx 9.4 \text{ feet}. \]

Step 3: Calculate the Area

To calculate the area of the circle, use the formula: \( A = \pi r^2 \). Substituting the given value for \( r \), we have:\[ A = \pi \times (1.5)^2 = \pi \times 2.25 \].With the approximation \( \pi \approx 3.14159 \), we find \( A \approx 3.14159 \times 2.25 = 7.0685775 \). Rounding this to the nearest tenth:\[ A \approx 7.1 \text{ square feet}. \]

Key Concepts

Understanding the Circumference Formula

The circumference of a circle represents the distance around the circle. Imagine taking a string and wrapping it around a circle; the length of that string would be the circumference.
To find this length, the circumference formula is used:

• Formula: \( C = 2 \pi r \)
• The symbol \( C \) stands for circumference, while \( \pi \) is a mathematical constant approximately equal to 3.14159. The variable \( r \) represents the radius, which is the distance from the center of the circle to any point on its perimeter.

To calculate the circumference:

• Identify the radius. For example, given \( r = 1.5 \) feet.
• Substitute the radius into the formula: \( C = 2 \pi \times 1.5 \).
• This results in \( C = 3 \pi \), which approximates to \( C \approx 3 \times 3.14159 \).
• After performing the multiplication, the result is rounded to the nearest tenth, giving \( C \approx 9.4 \) feet.

Exploring the Area Formula

The area of a circle is a measure of the space inside the circle. Imagine a filled-in circle; the area is the amount of material that would fill it.
To find the area, we use the area formula for circles:

• Formula: \( A = \pi r^2 \)
• Here, \( A \) stands for area, \( \pi \) is again the constant approximately equal to 3.14159, and \( r \) is the radius.

The steps to calculate the area:

• Use the given radius, such as \( r = 1.5 \) feet.
• Plug \( r \) into the formula: \( A = \pi \times (1.5)^2 \).
• Calculate \( (1.5)^2 \), which is 2.25.
• Multiply \( \pi \) by 2.25 to find \( A \approx 3.14159 \times 2.25 \).
• After computing, round the result to the nearest tenth: \( A \approx 7.1 \) square feet.

Understanding Pi Approximation

Pi, represented by the symbol \( \pi \), is a special mathematical constant. It is crucial in circle calculations because it represents the ratio of the circumference of any circle to its diameter.
Some important facts about \( \pi \):

• The exact value of \( \pi \) extends to infinite decimal places, approximately 3.14159.
• For everyday calculations, \( \pi \) is often rounded to 3.14, but using more decimal places like 3.14159 can increase accuracy.

When working with problems involving circles, using a precise value of \( \pi \) helps in obtaining a more accurate circumference or area. In most practical scenarios, rounding to 3.14159 provides a good balance between precision and simplicity.
Whenever you calculate measurements involving circles and \( \pi \), it's important to decide how accurate your answer needs to be and choose the suitable approximation of \( \pi \). This way, your results fit the level of accuracy necessary for the problem at hand.