Question

14 a circular clock has a minute hand that rotates from the center of the clock. this distance from the point on the minute hand from where it rotates to its end is 12 centimeters (cm). how far does the end of the minute hand move in 10 minutes?

Explanation:

Step1: Determine the angle rotated

A clock is a circle (360°), and the minute hand completes a full rotation (360°) in 60 minutes. So in 1 minute, it rotates \( \frac{360^\circ}{60} = 6^\circ \) per minute. In 10 minutes, it rotates \( 10\times6^\circ = 60^\circ \). To use the arc length formula, we convert this angle to radians. We know that \( 180^\circ=\pi \) radians, so \( 60^\circ = \frac{\pi}{3} \) radians (since \( 60^\circ=\frac{180^\circ}{3} \), so in radians \( \frac{\pi}{3} \)).

Step2: Recall the arc length formula

The formula for the length of an arc \( s \) is \( s = r\theta \), where \( r \) is the radius of the circle and \( \theta \) is the central angle in radians. Here, the radius \( r \) of the circular path of the minute hand is 12 cm (the length of the minute hand), and \( \theta=\frac{\pi}{3} \) radians.

Step3: Calculate the arc length

Substitute \( r = 12 \) and \( \theta=\frac{\pi}{3} \) into the arc length formula: \( s = 12\times\frac{\pi}{3} \). Simplifying this, \( 12\div3 = 4 \), so \( s = 4\pi \) cm. If we use \( \pi\approx3.14 \), then \( s\approx4\times3.14 = 12.56 \) cm. Alternatively, we can also think in terms of the fraction of the circumference. The circumference of a circle is \( C = 2\pi r \). The minute hand moves \( \frac{10}{60}=\frac{1}{6} \) of the full circumference. The circumference \( C = 2\pi\times12 = 24\pi \) cm. Then the arc length is \( \frac{1}{6}\times24\pi = 4\pi \approx12.56 \) cm.

Answer:

The end of the minute hand moves \( 4\pi \) centimeters (or approximately \( 12.56 \) centimeters).