A lighthouse, L, is on an island 4 mi away from the closest point, P, on the beach. If the lighthouse light rotates clockwise at a constant rate of 17 revolutions/min, how fast (in mi/min) does the beam of light move across the beach 2 mi away from the closest point on the beach?



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.