Chapter 6.1, Problem 90E

To determine

Calculate the radians.

Answer to Problem 90E

  $\frac{23\pi }{24}\approx 3.011rad$ and $d\approx 36.128rad$

Explanation of Solution

Given information:

In one hour, the minute hand on a clock moves through a complete circle, and the hour hand moves $\frac{1}{12}$ through of a circle. Calculate radians do the minute , hour hand move between $1:00p.m.$ and $6:45p.m$ (on the same day).

  

Calculation:

Consider the figures

  

In a standard wall clock, the minute hand completes a full circle and the hour and $1/12$ of a circle in one hour.

Now determine how many radians the hands move between $1:00p.m.\&6:45p.m$ on the same day as shown in the above figure.

First, the hour moves through an arc of $5 45/60or5 3/4hours.$

Because a complete revolution of a hand is $12hours$ ,the fraction of the clock face traversed by the hour hand is

  $\frac{5\frac{3}{4}h}{12h}=\frac{23}{48}$

Hence there are $2\pi$ radians in a full circle, the hour hand has travelled

  $\begin{array}{l}\frac{23}{48}\times 2\pi =\frac{23\pi }{24}\\ \frac{23\pi }{24}\approx 3.011rad\end{array}$

The minute hand has made five complete revolutions plus $3/4$ of a revolution.

Again multiply by $2\pi$ gives the distance, $d$ , travelled by the minute hand in radians.

  $\begin{array}{l}d=\left(5+\frac{3}{4}\right)2\pi \\ d=11.5\pi \\ d\approx 36.128rad\end{array}$