A circle has a central angle of \( \frac{5 \pi}{2} \) radians that intercepts an arc with a length of 46 inches. Based on this information, what is the measure of the radius of the circle? Round to the nearest hundredth (two decimal places).
Answer
5. Multiply both sides by \( \frac{2}{5 \pi} \) to get \( r \approx 5.85 \) inches.
Step 1 ：1. \( \frac{5 \pi}{2} \) radians represents a central angle of the circle, with an intercepted arc of 46 inches in length. We need to find the radius of the circle.
Step 2 ：2. Use the arc length formula: \( L = r \theta \), where \(L\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.
Step 3 ：3. Substitute the given values: \( 46 = r \left( \frac{5 \pi}{2} \right) \).
Step 4 ：4. Solve for \(r\): \( r = \frac{46}{\frac{5 \pi}{2}} \).
Step 5 ：5. Multiply both sides by \( \frac{2}{5 \pi} \) to get \( r \approx 5.85 \) inches.
