One a circle of radius 24 in, find the length of arc subtended by a central angle of (a) 2/3 rad, (b) 3 phi/5 rad,(c) 75°, (d) 130°

To find the length of an arc subtended by a central angle in a circle, we use the formula:

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

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

Given:  
- Radius \( r = 24 \) inches.

Let's find the arc lengths for each of the given angles:

### (a) \( \theta = \frac{2}{3} \) radians

\[
\text{Arc Length} = 24 \times \frac{2}{3} = 24 \times 0.6667 = 16 \text{ inches}
\]

### (b) \( \theta = \frac{3\pi}{5} \) radians

\[
\text{Arc Length} = 24 \times \frac{3\pi}{5} = 24 \times 1.884 = 45.21 \text{ inches} 
\]

### (c) \( \theta = 75^\circ \)

First, convert degrees to radians:

\[
\theta = 75^\circ \times \frac{\pi}{180^\circ} = \frac{75\pi}{180} = \frac{5\pi}{12} \text{ radians}
\]

Then, calculate the arc length:

\[
\text{Arc Length} = 24 \times \frac{5\pi}{12} = 24 \times 1.309 = 31.42 \text{ inches}
\]

### (d) \( \theta = 130^\circ \)

Convert degrees to radians:

\[
\theta = 130^\circ \times \frac{\pi}{180^\circ} = \frac{130\pi}{180} = \frac{13\pi}{18} \text{ radians}
\]

Then, calculate the arc length:

\[
\text{Arc Length} = 24 \times \frac{13\pi}{18} = 24 \times 2.268 = 54.43 \text{ inches}
\]

### Summary of Results:
- (a) \( \frac{2}{3} \) radians: **16 inches**
- (b) \( \frac{3\pi}{5} \) radians: **45.21 inches**
- (c) \( 75^\circ \): **31.42 inches**
- (d) \( 130^\circ \): **54.43 inches**

Would you like more details on any of these steps or have any other questions?

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**Related Questions:**

1. How do you convert radians to degrees and vice versa?
2. What is the formula for the circumference of a circle?
3. How do you calculate the area of a sector of a circle?
4. What is the difference between a major arc and a minor arc?
5. How do you find the length of an arc when the central angle is given in degrees?

**Tip:** Remember, when converting between degrees and radians, \( 180^\circ = \pi \) radians.

Learn how to calculate the arc length on a circle with a radius of 24 inches using various central angles in radians and degrees. Solve for angles including 2/3 radians, 3pi/5 radians, 75 degrees, and 130 degrees with step-by-step solutions.

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