To solve the problem of finding the circumference of the circle given that a central angle measuring [tex]\(\frac{3\pi}{4}\)[/tex] radians intersects an arc of length 45 inches, follow these steps:
1. Understand the relationship between arc length, radius, and central angle:
- The formula that relates arc length ([tex]\(L\)[/tex]), radius ([tex]\(r\)[/tex]), and central angle ([tex]\(\theta\)[/tex]) in radians is:
[tex]\[
L = r \theta
\][/tex]
2. Given values:
- Central angle [tex]\(\theta = \frac{3\pi}{4}\)[/tex] radians
- Arc length [tex]\(L = 45\)[/tex] inches
3. Calculate the radius (r) of the circle:
- Rearrange the formula for the arc length to solve for the radius:
[tex]\[
r = \frac{L}{\theta}
\][/tex]
- Substitute the given values into the formula:
[tex]\[
r = \frac{45}{\frac{3\pi}{4}}
\][/tex]
- Numerical substitution using [tex]\(\pi \approx 3.14\)[/tex]:
[tex]\[
r = \frac{45}{\frac{3 \times 3.14}{4}} = \frac{45}{2.355} \approx 19.10828025477707 \text{ inches}
\][/tex]
4. Calculate the circumference of the circle:
- The formula for the circumference ([tex]\(C\)[/tex]) of a circle is:
[tex]\[
C = 2 \pi r
\][/tex]
- Substitute the radius and [tex]\(\pi\)[/tex] into the formula:
[tex]\[
C = 2 \times 3.14 \times 19.10828025477707 \approx 120.00000000000001 \text{ inches}
\][/tex]
5. Round the circumference to the nearest tenth:
- Rounded circumference = 120.0 inches
The circumference of the circle is 120.0 inches, rounded to the nearest tenth.
