Answer :
Answer:
34[tex]\pi[/tex] miles per minute
Step-by-step explanation:
Let's denote:
- The distance between the lighthouse and point P on the beach as
[tex]d_{1}[/tex] = 2 miles
- The distance from point P to the location 2 miles away on the beach as [tex]d_{2}[/tex]= 2 miles
The rate at which the lighthouse light rotates can be considered as the angular velocity of the light beam. Given that the light rotates clockwise at a constant rate of 17 revolutions per minute, we can calculate the angular velocity:
The circumference of the circle formed by the light beam is 2[tex]\pi[/tex][tex]d_{1}[/tex] miles. As the lighthouse light makes 17 revolutions per minute, the speed of the beam of light moving across the beach at point P is:
Speed = 17 x 2[tex]\pi[/tex][tex]d_{1}[/tex] miles/min
Now, let's calculate the linear speed of the light beam at the location 2 miles away on the beach:
At a distance of 2 miles from point P, the beam of light is moving across the beach at a distance of 2 miles from the point of contact. This creates a smaller circle with a circumference of 2[tex]\pi[/tex]x2 miles.
Therefore, the speed of the light beam at a location 2 miles away on the beach is:
speed at miles away: 17 x 2[tex]\pi[/tex] x 4 x [tex]\frac{2}{4}[/tex] miles/min
speed at miles away: 17 x 2[tex]\pi[/tex] x 2 miles/min
speed at miles away: 34[tex]\pi[/tex] miles/min
Hence, the speed at which the light beam moves across the beach 2 miles away from the closest point on the beach is 34[tex]\pi[/tex] miles per minute.
