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?

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.