Question

If the diameter of a circle is 30 inches, and a central angle of 7π4 creates a minor arc, how long is that arc? Angle is in radians.

(1 point)
Responses

52.5 in.
52.5 in.

82.5 in.
82.5 in.

49 in.
49 in.

26.25 in.

Answer 1

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

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

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

1. **Calculate the radius**: Given that the diameter of the circle is 30 inches:
\[
r = \frac{\text{diameter}}{2} = \frac{30}{2} = 15 \text{ inches}
\]

2. **Use the given central angle**: The angle \( \theta \) is given as \( \frac{7\pi}{4} \) radians.

3. **Substitute the values into the formula**:
\[
\text{Arc Length} = r \cdot \theta = 15 \cdot \frac{7\pi}{4}
\]

4. **Calculate the arc length**:
\[
\text{Arc Length} = 15 \cdot \frac{7\pi}{4} = \frac{105\pi}{4}
\]

5. **Now compute \( \frac{105\pi}{4} \) numerically** (using \( \pi \approx 3.14 \)):
\[
\frac{105 \cdot 3.14}{4} \approx \frac{329.7}{4} \approx 82.425 \text{ inches}
\]

Thus, the arc length is approximately **82.5 inches**.

The answer is **82.5 in.**