Question

If the diameter of a circle is 30 inches and a central angle of
7
π
4
creates a minor arc, then how long is that arc?(1 point)
Responses

49 in.
49 in.

52.5 in.
52.5 in.

82.5 in.
82.5 in.

26.25 in.

Answer 1

To find the length of an arc created by a central angle in a circle, you can use the formula:

\[
\text{Arc Length} = r \cdot \theta
\]

where:
- \( r \) is the radius of the circle,
- \( \theta \) is the angle in radians.

1. First, find the radius of the circle. The radius is half of the diameter:

\[
r = \frac{d}{2} = \frac{30 \text{ inches}}{2} = 15 \text{ inches}
\]

2. The central angle given is \( \frac{7\pi}{4} \) radians.

3. Now, use the arc length formula:

\[
\text{Arc Length} = 15 \text{ inches} \cdot \frac{7\pi}{4}
\]

To calculate this:

\[
\text{Arc Length} = 15 \cdot \frac{7\pi}{4} = \frac{105\pi}{4} \text{ inches}
\]

Now, substituting \( \pi \approx 3.14 \) to get a numerical approximation:

\[
\text{Arc Length} \approx \frac{105 \cdot 3.14}{4} \approx \frac{329.7}{4} \approx 82.425 \text{ inches}
\]

Thus, the arc length is approximately \( 82.5 \) inches.

The answer is **82.5 in.**