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.)

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:

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

- Distance along the straight shoreline, in kilometers.

- Angle of rotation of the lighthouse, in sexagesimal degrees.

To find the rate of change of distance along the straight shoreline (), in kilometers per minute, we use the following trigonometric relationship:

(1)

Then, we differentiate this expression in time:

(2)

Where 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): ,

If we know that , and , then the rate of change is:

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