A circle has a radius of 4 in . Find the length 5 of the arc intercepted by a central angle of \( 73^{\circ} \).

To find the length of the arc intercepted by a central angle, we can use the formula: \[ \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r \] where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle. Here, the radius \(r = 4\) inches and \(\theta = 73^\circ\). First, we calculate the circumference of the circle: \[ 2\pi r = 2\pi \times 4 = 8\pi \text{ inches} \] Now substitute the values into the arc length formula: \[ \text{Arc Length} = \frac{73}{360} \times 8\pi \] Calculating that gives: \[ \text{Arc Length} = \frac{73 \times 8\pi}{360} = \frac{584\pi}{360} \] Simplifying \(\frac{584}{360}\): \[ \frac{584}{360} = \frac{146}{90} = \frac{73}{45} \] So, the arc length is: \[ \text{Arc Length} = \frac{73\pi}{45} \text{ inches} \] This is approximately \(5.1\) inches when calculated with \(\pi \approx 3.14\). Thus, the length of the arc intercepted by a central angle of \(73^\circ\) is \(\frac{73\pi}{45}\) inches or approximately \(5.1\) inches.