Chapter 4.1, Problem 38E

To determine

To find: The speed of the beam of light moving along the shoreline.

Answer to Problem 38E

The speed of the moving light is $\frac{80\pi }{3}$ kilometers/minute.

Explanation of Solution

Given information:

The given distance from island to point $P$ is $3$ km, and light makes four revolution per minute.

Calculation:

Consider the figure.

  

Figure $\left(1\right)$

In the figure, look at the angle $\theta$ .

If light makes four revolution per minute.

Thus,

  $\begin{array}{c}\frac{d\theta }{dt}=\frac{4\times 2\pi \text{ Radians}}{1\text{ minute}}\\ =8\pi \text{ Radian/minute}\end{array}$

Thus,

  $\tan \theta =\frac{x}{3}$

Differentiate both sides with respect to $t$ .

  $\begin{array}{l}\frac{d\left(\tan \theta \right)dt}{dt}=\frac{1}{3}\cdot \frac{dx}{dt}\\ {\sec }^2\theta \cdot \frac{d\theta }{dt}=\frac{1}{3}\cdot \frac{dx}{dt}\end{array}$

When $x=1$ , $\tan \theta =\frac{1}{3}$ , and $\frac{dx}{dt}=8\pi$ .

  $\begin{array}{c}\left(1+\frac{1}{9}\right)\cdot \frac{d\theta }{dt}=\frac{1}{3}\cdot \frac{dx}{dt}\\ \frac{10}{9}.8\pi =\frac{1}{3}.\frac{dx}{dt}\\ \frac{dx}{dt}=\frac{80\pi }{3}\end{array}$

Therefore the speed of the moving light is $\frac{80\pi }{3}$ kilometers/minute.