Question

through how many radians will the hour hand on a clock rotate in (a) 13 hr? in (b) 6 hr?
(a) in 13 hr, the hour hand on the clock will rotate □ radians.
(type an exact answer in terms of π. type an integer or a simplified fraction.)
(b) in 6 hr, the hour hand on the clock will rotate □ radians.
(type an exact answer in terms of π. type an integer or a simplified fraction.)

Explanation:

Step1: Recall the formula for the rotation of the hour hand

The hour hand of a clock completes a full - circle (\(2\pi\) radians) in 12 hours. The formula for the angle \(\theta\) (in radians) rotated by the hour hand in \(t\) hours is \(\theta=\frac{2\pi}{12}t=\frac{\pi}{6}t\).

Step2: Calculate for \(t = 13\) hours

Substitute \(t = 13\) into the formula \(\theta=\frac{\pi}{6}t\).
\(\theta=\frac{\pi}{6}\times13=\frac{13\pi}{6}\) radians.

Step3: Calculate for \(t = 6\) hours

Substitute \(t = 6\) into the formula \(\theta=\frac{\pi}{6}t\).
\(\theta=\frac{\pi}{6}\times6=\pi\) radians.

Answer:

(a) \(\frac{13\pi}{6}\) radians; (b) \(\pi\) radians