Question

If the diameter of a circle is 30 inches and a central angle of 7π4 creates a minor arc, then how long is that arc?(1 point)
Responses

82.5 in.
82.5 in.

49 in.
49 in.

26.25 in.
26.25 in.

52.5 in.

Answer 1

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

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

where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.

Given:
- The diameter of the circle is 30 inches, so the radius \(r\) is:
\[
r = \frac{30}{2} = 15 \text{ inches}
\]

- The central angle is \(7\pi/4\) radians.

Now, we can substitute the values into the arc length formula:

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

We can calculate this step by step:

1. Calculate \(15 \times 7 = 105\).
2. Then divide by 4:
\[
\frac{105\pi}{4}
\]
3. To approximate it numerically, use \(\pi \approx 3.14\):
\[
\frac{105 \times 3.14}{4} \approx \frac{329.7}{4} \approx 82.425 \text{ inches}
\]

Therefore, the length of the arc is approximately 82.5 inches.

So the answer is **82.5 in.**