Question

A lighthouse is located on a small island 5 km away from the nearest point P on a straight shoreline and its light makes six revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? (Round your answer to one decimal place.)

Answer 1

Answer:

The beam of light is moving along the shoreline at a velocity of approximately 196 kilometers per second.

Step-by-step explanation:

The statement is described geometrically in the image attached below by means of a right triangle. All variables are described below:

$OP$ - Minimum distance between lighthouse and the straight shoreline, in kilometers.

$PP'$ - Distance along the straight shoreline, in kilometers.

$\theta$ - Angle of rotation of the lighthouse, in sexagesimal degrees.

To find the rate of change of distance along the straight shoreline ($\frac{dPP'}{dt}$), in kilometers per minute, we use the following trigonometric relationship:

$PP' = OP \cdot \tan \theta$ (1)

Then, we differentiate this expression in time:

$\frac{dPP'}{dt} = OP\cdot \dot \theta \cdot \sec^{2}\theta$ (2)

Where $\dot \theta$ is the rate of change of the angle of rotation of the lighthouse, in radians per minute.

The angle at the given instant is calculated by (1): $OP = 5\,km$, $PP' = 1\,km$

$\theta = \tan^{-1} \left(\frac{PP'}{OP} \right)$

$\theta \approx 11.310^{\circ}$

If we know that $\dot \theta = 37.699\,\frac{rad}{min}$, $\theta \approx 11.310^{\circ}$ and $OP = 5\,km$, then the rate of change is:

$\frac{dPP'}{dt} \approx 196,035\,\frac{km}{min}$

The beam of light is moving along the shoreline at a velocity of approximately 196 kilometers per second.