Q8:The minute hand of a clock is 5 cm long and moves from 12 to 4 o'clock. This is \( \frac{1}{3} \) of a complete revolution. a. Draw a picture to show this and give the measure of the angle in degrees that the hand moves? b. Through how many radians does it move? c. How far (in cm) does the tip of the minute hand move from 12 to 4 o'clock? Express your answer in terms of \( \pi \).

To solve the problem, let's break it down into parts: ### a. Drawing and Angle Measure 1. **Drawing**: You can visualize a clock face. The minute hand starts at the 12 o'clock position and moves to the 4 o'clock position. This movement covers 4 hour marks (12 to 1, 1 to 2, 2 to 3, and 3 to 4). 2. **Angle Measure**: - A complete revolution of the clock is \(360^\circ\). - Since the minute hand moves from 12 to 4, it covers 4 hour marks. - Each hour mark represents \( \frac{360^\circ}{12} = 30^\circ\). - Therefore, the angle the minute hand moves is: \[ 4 \times 30^\circ = 120^\circ \] ### b. Radians To convert degrees to radians, we use the conversion factor \( \frac{\pi \text{ radians}}{180^\circ} \): \[ \text{Angle in radians} = 120^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{120\pi}{180} = \frac{2\pi}{3} \text{ radians} \] ### c. Distance Traveled by the Tip of the Minute Hand The distance traveled by the tip of the minute hand can be calculated using the formula for the arc length: \[ \text{Arc length} = r \theta \] where \(r\) is the radius (length of the minute hand) and \(\theta\) is the angle in radians. 1. The length of the minute hand \(r = 5 \text{ cm}\). 2. The angle in radians \(\theta = \frac{2\pi}{3}\). Now, substituting these values into the arc length formula: \[ \text{Arc length} = 5 \times \frac{2\pi}{3} = \frac{10\pi}{3} \text{ cm} \] ### Summary of Answers a. The angle the minute hand moves is \(120^\circ\). b. The angle in radians is \(\frac{2\pi}{3}\). c. The distance the tip of the minute hand moves is \(\frac{10\pi}{3} \text{ cm}\).