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:
[tex]OP[/tex] - Minimum distance between lighthouse and the straight shoreline, in kilometers.
[tex]PP'[/tex] - Distance along the straight shoreline, in kilometers.
[tex]\theta[/tex] - Angle of rotation of the lighthouse, in sexagesimal degrees.
To find the rate of change of distance along the straight shoreline ([tex]\frac{dPP'}{dt}[/tex]), in kilometers per minute, we use the following trigonometric relationship:
[tex]PP' = OP \cdot \tan \theta[/tex] (1)
Then, we differentiate this expression in time:
[tex]\frac{dPP'}{dt} = OP\cdot \dot \theta \cdot \sec^{2}\theta[/tex] (2)
Where [tex]\dot \theta[/tex] 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): [tex]OP = 5\,km[/tex], [tex]PP' = 1\,km[/tex]
[tex]\theta = \tan^{-1} \left(\frac{PP'}{OP} \right)[/tex]
[tex]\theta \approx 11.310^{\circ}[/tex]
If we know that [tex]\dot \theta = 37.699\,\frac{rad}{min}[/tex], [tex]\theta \approx 11.310^{\circ}[/tex] and [tex]OP = 5\,km[/tex], then the rate of change is:
[tex]\frac{dPP'}{dt} \approx 196,035\,\frac{km}{min}[/tex]
The beam of light is moving along the shoreline at a velocity of approximately 196 kilometers per second.
