Chapter 9.2, Problem 50PS

Solution

To find: The measure of angle generated by hour hand of a clock from $9:00\text{A}\text{.M}\text{.}$ to $5:00\text{P}\text{.M}\text{.}$ in both degrees and radians.

The angle made by the hour hand is $-{240}^{\circ }$ or $-4.19\text{radians}$ .

The angle made by the minute hand is $12$ times more than the angle made by the hour hand.

Given information:

A person works every Saturday from $9:00\text{A}\text{.M}\text{.}$ to $5:00\text{P}\text{.M}\text{.}$ Draw a diagram that shows the rotation completed by the hour hand of a clock during this time. Find the measure of the angle generated by the hour hand in both degrees and radians. Compare this angle with the angle generated by the minute hand from $9:00\text{A}\text{.M}\text{.}$ to $5:00\text{P}\text{.M}\text{.}$

Explanation:

The diagram below shows the rotation completed by the hour hand of a clock during this time:

  

There are ${360}^{\circ }$ in one complete rotation of twelve hours by the hour hand.

So, one hour has:

  $\frac{{360}^{\circ }}{12}={30}^{\circ }$

Since there are $8$ hours from $9:00\text{A}\text{.M}\text{.}$ to $5:00\text{P}\text{.M}\text{.}$, it follows that:

The hour hand will cover:

  $30\times 8={240}^{\circ }$

The rotation is clockwise so the angle covered by the hour hand will be $-{240}^{\circ }$ .

Multiply the value by $\frac{\prod }{180}$ to convert the angle into radians:

  $\begin{array}{l}-240\times \frac{\prod }{180}\\ =-240\times \frac{3.14}{180}\\ \approx -4.19\end{array}$

Hence, the value is $-4.19\text{radians}$ approximately.

The minute hand makes a complete revolution in each hour.

Therefore, the revolutions made by the minute hand in $8$ hours from $9:00\text{A}\text{.M}\text{.}$ to $5:00\text{P}\text{.M}\text{.}$ will be:

  $8\times \left(-{360}^{\circ }\right)=-{2880}^{\circ }$

Now, compare this angle with the angle made by the hour hand:

  $\frac{-{2880}^{\circ }}{-{240}^{\circ }}=12$

Hence, the angle made by the minute hand is $12$ times more than the angle made by the hour hand.