Q. 91. - Precalculus Enhanced with Graphing Utilities

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. 

Step-by-Step Solution

Answer

The tip of the minute hand moves$9.42$ inches in$15$  minutes.

It moves $15.71$inches in$25$ minutes.

Step 1. Given Information

The minute hand of a clock is $6$ inches long. 

Step 2. Intervals of the clock

First we need to know the intervals of the clock by converting to the minute intervals. 

One revolution of the clock hand is $2\mathrm{\pi }$ radians.

Also, $1$ hour = $60$ minutes.

So, each minute indicates, $\frac{2\mathrm{\pi }}{60}=\frac{\mathrm{\pi }}{30}$ radians.

Step 3. Consideration

The length of the minute hand which is $6$ inches is considered as the radius and the first $15$ minutes is the arc length of a circle.

Step 4. Calculation for 15 minutes

To solve for the arc length in $15$ minutes, we use the formula $S_{15}=r\theta$, where$r=6$ inches and $\theta =15\left(\frac{\mathrm{\pi }}{30}\right)$

Therefore,  $$\begin{gathered} S_{15}=6·15·\left(\frac{\mathrm{\pi }}{30}\right) \\ {S}_{15}=\frac{90\mathrm{\pi }}{30}=3\mathrm{\pi }\approx 9.42 \end{gathered}$$ inches.

Step 5. Calculation for 25 minutes

To solve for the arc length of $25$ minutes, we use the formula,$S_{25}=r\theta$ where $r=6$ inches and $\theta =25·\left(\frac{\mathrm{\pi }}{30}\right)$