Problem 17 - Basic Technical Mathematics with Calculus

Question

Find the area of the circle with the given circumference. $$c=40.1 \mathrm{cm}$$

Step-by-Step Solution

Answer

The area of the circle is approximately 127.8 cm².

Step 1: Recall the formula for the circumference of a circle

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

Step 2: Solve for the radius

Set \( C = 40.1 \) cm in the circumference formula to solve for \( r \):\[ 40.1 = 2\pi r \]Divide both sides by \( 2\pi \) to isolate \( r \):\[ r = \frac{40.1}{2\pi} \].

Step 3: Calculate the radius

Substitute \( \pi \approx 3.1416 \) to find the numerical value of \( r \):\[ r = \frac{40.1}{2 \times 3.1416} \approx \frac{40.1}{6.2832} \approx 6.38 \text{ cm} \].

Step 4: Recall the formula for the area of a circle

The area of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle. Use this formula to find the area.

Step 5: Calculate the area of the circle

Substitute the radius \( r \approx 6.38 \) cm into the area formula:\[ A = \pi (6.38)^2 \approx 3.1416 \times 40.7044 \approx 127.8 \text{ cm}^2 \].

Key Concepts

Circumference of a Circle

The circumference of a circle is essentially the distance around the circle's edge. Think of it like the perimeter of a circle. The formula used to calculate the circumference, which is an important part of any circle-related problem, is given by:
\[C = 2\pi r\]where:

• \( C \) represents the circumference,
• \( \pi \) is a constant approximately equal to 3.1416,
• \( r \) is the radius of the circle.

This formula tells us that the circumference is dependent on the radius of the circle. By knowing the circumference, we can also work backwards to find out the radius, as shown in the problem with the given circumference of 40.1 cm.
Breaking down the circumference formula can give insight into how the circle's size is quantified in terms of its radius and \( \pi \). Understanding this relationship helps with visualizing and solving more complex geometric problems where the circumference might not be directly given.

Radius Calculation Formula

Finding the radius from a given circle measurement is a common step when dealing with problems involving circles. When the circumference is known, we can rearrange the circumference formula to solve for the radius \( r \):
\[C = 2\pi r \]Rearranging gives:\[r = \frac{C}{2\pi}\]This formula essentially allows us to decompose the total distance around the circle (the circumference) in terms of the radius and the constant \( \pi \).
For the exercise's example, by plugging in the given value \( C = 40.1 \, cm \) into this formula, and using \( \pi \approx 3.1416 \), we find:
\[r \approx \frac{40.1}{6.2832} \approx 6.38 \, cm\]Here, we've divided the circumference value by twice the value of \( \pi \), yielding the radius. It's always crucial to ensure that the units used in these calculations are consistent and correct.

Area of a Circle Formula

Once you have the radius, you can easily calculate the area of the circle. The formula for the area uses both the radius and \( \pi \):
\[A = \pi r^2\]where:

• \( A \) stands for area,
• \( r \) is the radius of the circle,
• \( \pi \) remains approximately 3.1416.

The formula \( A = \pi r^2 \) shows us that the area grows with the square of the radius. This means that even a small increase in radius will result in a larger increase in area.
In the given exercise, with \( r \approx 6.38 \, cm \), we substitute back into the area formula to find:
\[A \approx 3.1416 \times (6.38)^2 \approx 3.1416 \times 40.7044 \approx 127.8 \, cm^2\]This calculation combines both geometric and arithmetic skills, highlighting how circle properties are interrelated. Mastering this formula opens doors to solving various real-world measurement problems, from land area computations to determining the materials needed for circular decorations.