Question

The minute hand of a clock is 6 inches long. how far does the tip of the minute hand move in 15 minutes? how far does it move in 25 minutes? round answers to two decimal places.

Answer 1

This problem involves finding arc lengths.  The formula for arc length 

is    s = r*theta, where r is the radius and theta the central angle.

In 15 minutes, the minute hand sweeps out 1/4 of a circle, or pi/2 radians.  This is the central angle.  The arc length (how far the minute hand moves in 15 min) is then

s = (6 inches)(pi/2 rad) = 3pi inches, or about 9.42 inches.

25 minutes is equivalent to a central angle of (25/60)pi rad, or 1.31 radians.  What is the associated arc length?  Calculate this in the same way as I did for a central angle of pi/2.

Answer 2

Answer:

After 15 minutes it moves 9.42 inches, and after 35 minutes it moves 15.7 inches.

Step-by-step explanation:

For this problem we use the following formula

$s=r\theta$

Where $s$ is the arc, $r$ is the radius and $\theta$ is the angle that subtends that arc.

In this case, we know that $r=6in$ and $\theta =90\°$ where it moves for 15 minutes, because each 5 minutes, the minute man moves $\frac{360\°}{12}=30\°$.

Now, if we transform from degrees to radians, it would be

$90\°\frac{\pi}{180\°}=\frac{\pi}{2}$

Using $\pi \approx 3.14$, the distance moved after 15 minutes would be

$s=r\theta=6\frac{\pi}{2}= 3 \pi \approx 9.42 \ in$

Then, we do the same process when it moves for 25 minutes. As we said before, the minute hand moves 30° each 5 minutes. So, after 25 minutes it moves 150°.

Transforming from degrees to radians, we have

$150\°\frac{\pi}{180\°}= \frac{5 \pi}{6}$

Then, we replace in the first formula

$s=r\theta=6(\frac{5\pi}{6})=5\pi \approx 15.7 \ in$

Therefore, after 15 minutes it moves 9.42 inches, and after 35 minutes it moves 15.7 inches.