A circle with a \( \frac{4 \pi}{3} \) radian central angle has a radius of 15 feet. 1. Find the length of the arc intercepted by the central angle 2. Express the measure of the angle in degrees. 3. Find the circumference of the circle. Use 3.14 to approximate pi in your calculation. For full cradit wou muer.

Let's solve the problem step by step. ### Step 1: Find the length of the arc intercepted by the central angle The formula for the length of an arc \( L \) in a circle is given by: \[ L = r \theta \] where: - \( r \) is the radius of the circle, - \( \theta \) is the central angle in radians. Given: - \( r = 15 \) feet, - \( \theta = \frac{4 \pi}{3} \) radians. Now, we can calculate the length of the arc: \[ L = 15 \times \frac{4 \pi}{3} \] ### Step 2: Express the measure of the angle in degrees To convert radians to degrees, we use the formula: \[ \text{Degrees} = \theta \times \frac{180}{\pi} \] Substituting \( \theta = \frac{4 \pi}{3} \): \[ \text{Degrees} = \frac{4 \pi}{3} \times \frac{180}{\pi} \] ### Step 3: Find the circumference of the circle The formula for the circumference \( C \) of a circle is: \[ C = 2 \pi r \] Using \( r = 15 \) feet and approximating \( \pi \) as 3.14: \[ C = 2 \times 3.14 \times 15 \] Now, let's perform the calculations for all three parts. Calculate the value by following steps: - step0: Calculate: \(2\times 3.14\times 15\) - step1: Multiply the terms: \(6.28\times 15\) - step2: Multiply the numbers: \(94.2\) Calculate or simplify the expression \( (4 * 180 / 3) \). Calculate the value by following steps: - step0: Calculate: \(\frac{4\times 180}{3}\) - step1: Multiply the numbers: \(\frac{720}{3}\) - step2: Reduce the fraction: \(240\) Calculate or simplify the expression \( 15 * (4 * 3.14 / 3) \). Calculate the value by following steps: - step0: Calculate: \(15\left(\frac{4\times 3.14}{3}\right)\) - step1: Multiply the numbers: \(15\left(\frac{12.56}{3}\right)\) - step2: Divide the terms: \(15\times \frac{314}{75}\) - step3: Reduce the numbers: \(1\times \frac{314}{5}\) - step4: Multiply: \(\frac{314}{5}\) Here are the results for each part of the problem: ### 1. Length of the arc intercepted by the central angle The length of the arc \( L \) is approximately: \[ L \approx 62.8 \text{ feet} \] ### 2. Measure of the angle in degrees The measure of the angle in degrees is: \[ \text{Degrees} = 240^\circ \] ### 3. Circumference of the circle The circumference \( C \) of the circle is approximately: \[ C \approx 94.2 \text{ feet} \] ### Summary of Results - Length of the arc: \( 62.8 \) feet - Angle in degrees: \( 240^\circ \) - Circumference: \( 94.2 \) feet If you have any further questions or need additional assistance, feel free to ask!