Problem 18 - Basic Technical Mathematics with Calculus

Question

Find the area of the circle with the given circumference. $$c=147 \mathrm{m}$$

Step-by-Step Solution

Answer

The area of the circle is approximately 1719.69 square meters.

Step 1: Identify the Formula for Circumference

The formula to find the circumference of a circle is given by \( C = 2 \pi r \), where \( C \) is the circumference and \( r \) is the radius of the circle.

Step 2: Solve for the Radius

Using the formula \( C = 2 \pi r \), we can solve for the radius \( r \). Substitute the given circumference \( C = 147 \) into the formula:\[ 147 = 2 \pi r \]Now solve for \( r \):\[ r = \frac{147}{2\pi} \]

Step 3: Substitute the Radius in the Area Formula

The area \( A \) of the circle is given by \( A = \pi r^2 \). Use the value of \( r \) found in Step 2:\[ r = \frac{147}{2\pi} \]So, the area is:\[ A = \pi \left(\frac{147}{2\pi}\right)^2 \]

Step 4: Simplify the Area Expression

Simplify:\[ A = \pi \times \left(\frac{147^2}{(2\pi)^2}\right) \]\[ A = \frac{147^2}{4\pi} \]Now calculate \( 147^2 \):\( 147^2 = 21609 \)Substitute back:\[ A = \frac{21609}{4\pi} \]

Step 5: Calculate the Area Numerically

Finally, numerically evaluate the area. Using \( \pi \approx 3.14159 \):\[ A = \frac{21609}{4 \times 3.14159} \]\[ A \approx \frac{21609}{12.56636} \]\[ A \approx 1719.689 \]The area of the circle is approximately \( 1719.69 \) square meters.

Key Concepts

Circumference of a Circle

The circumference of a circle is the distance around the circle. Think of it like the perimeter you would walk if you were going all the way around the edge. It's a crucial concept in geometry, especially when solving problems involving circular shapes.
The formula for finding the circumference is:

• \( C = 2 \pi r \)

In this formula:

• \( C \) represents the circumference.
• \( \pi \) is a constant approximately equal to 3.14159.
• \( r \) is the radius of the circle.

To understand this better, consider you know the circumference of a circle, as in our problem where \( C = 147 \) meters. By knowing \( C \), you can find other important measurements, like the radius, which can lead to calculating the area of the circle.

Area of a Circle

The area of a circle is the space contained within its boundaries. Imagine painting inside the circle or cutting out a circular piece of paper; the amount of paint or paper you use is what we call the area.
The formula to calculate the area is:

• \( A = \pi r^2 \)

Where:

• \( A \) is the area.
• \( \pi \) is approximately 3.14159, just like before.
• \( r \) is the radius again.

In the given problem, we've first calculated the radius using the given circumference, then applied the radius to this formula to find the area. This shows how intertwined the concepts of circumference, radius, and area are for circle-related problems. Once all formulas and calculations are set, simplifying the expression leads to a numerical result that offers insight into the circle's size.

Radius of a Circle

The radius is an essential part of understanding a circle. It is the straight-line distance from the center of the circle to any point on its edge. Knowing the radius helps in understanding both the circumference and the area.
When you have the circle's circumference, you can find the radius by rearranging the circumference formula:

• \( C = 2 \pi r \)

Solving for the radius gives:

• \( r = \frac{C}{2\pi} \)

So, if you know \( C = 147 \) meters, you plug that number into the formula to get:

• \( r = \frac{147}{2\pi} \)

Finding \( r \) is crucial because it serves as the foundation for calculating other properties of the circle, like area. This equation illustrates how starting with one measurement, like the circumference, allows you to solve for other critical characteristics like the radius, ultimately making the circle's properties more accessible and understandable.