Question

1. a circle has radius 20 meters. a sector has a central angle of 45°.

a) find the arc length of the sector
b) find the area of the sector

1. the minute hand of a clock is 9 inches long. how far does the tip of the minute hand travel in 20 minutes?

hint: what fraction of the full circle does 20 minutes represent?

Explanation:

Problem 10a:

Step1: Convert angle to radians

The central angle is \(45^\circ\). To convert to radians, use the formula \( \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \). So, \( 45^\circ \times \frac{\pi}{180} = \frac{\pi}{4} \) radians.

Step2: Use arc length formula

The arc length formula is \( s = r\theta \), where \( r = 20 \) meters and \( \theta = \frac{\pi}{4} \). Substitute the values: \( s = 20 \times \frac{\pi}{4} = 5\pi \approx 15.71 \) meters.

Step1: Recall sector area formula

The formula for the area of a sector is \( A = \frac{1}{2}r^2\theta \), where \( r = 20 \) meters and \( \theta = \frac{\pi}{4} \) radians (from part a).

Step2: Substitute values

Substitute \( r = 20 \) and \( \theta = \frac{\pi}{4} \) into the formula: \( A = \frac{1}{2} \times 20^2 \times \frac{\pi}{4} \). Calculate \( 20^2 = 400 \), then \( \frac{1}{2} \times 400 = 200 \), and \( 200 \times \frac{\pi}{4} = 50\pi \approx 157.08 \) square meters.

Step1: Determine fraction of circle

A clock has 60 minutes for a full circle (360°). In 20 minutes, the fraction of the circle is \( \frac{20}{60} = \frac{1}{3} \).

Step2: Calculate circumference

The circumference of a circle is \( C = 2\pi r \), where \( r = 9 \) inches. So, \( C = 2\pi \times 9 = 18\pi \) inches.

Step3: Find distance traveled

The distance traveled is the fraction of the circumference: \( \frac{1}{3} \times 18\pi = 6\pi \approx 18.85 \) inches.

Answer:

The arc length is \( 5\pi \) meters (or approximately \( 15.71 \) meters).

Problem 10b: