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.