a lighthouse is 1 km from the shoreline and revolves at 10 pi radians a minute what is the speed of the light as it sweeps across the shore as it lights up at sand at 2 km from the light house

wow -- a lighthouse that rotates! usually it's just the light!

On shore, consider the lighted sand that is x km from the spot on shore nearest the lighthouse.

tanθ = x
sec^2θ dθ/dt = dx/dt
4*10pi = dx/dt
dx/dt = 40pi km/min

To find the speed of the light as it sweeps across the shore, we need to calculate the linear speed at which the light moves.

The circumference of a circle with a radius of 1 km is given by the formula C = 2πr, where r is the radius. Plugging in the value, we get C = 2π(1) = 2π km.

Since the lighthouse revolves at 10π radians per minute, we need to convert this angular speed to linear speed. The formula to convert angular speed to linear speed is:

Linear speed = Angular speed * Radius

Since the radius is 1 km, the linear speed is:

Linear speed = 10π * 1 = 10π km/min

Therefore, the speed of the light as it sweeps across the shore is 10π km/min.

To find the speed of the light as it sweeps across the shore, we need to calculate the linear speed of the light. This can be done by multiplying the angular velocity (radians per minute) by the distance from the lighthouse to the point on the shoreline where the light is directed.

In this case, the angular velocity is given as 10 pi radians per minute, and the point where the light hits the shore is 2 km from the lighthouse. Since the lighthouse itself is 1 km from the shoreline, the distance from the lighthouse to the point where the light hits the shore is 1 km + 2 km = 3 km.

To convert to meters and minutes for consistency, we need to convert the distance from kilometers to meters and the angular velocity from radians per minute to radians per second.

Given that 1 kilometer is equal to 1000 meters, the distance is 3 km * 1000 m/km = 3000 meters.

To convert 10 pi radians per minute to radians per second, we divide by 60 (the number of seconds in a minute). 10 pi radians/minute / 60 seconds/minute = (10 pi / 60) radians/second.

Finally, to calculate the linear speed, we multiply the angular velocity by the distance: (10 pi / 60) radians/second * 3000 meters = (5 pi / 3) meters/second.

Therefore, the speed of the light as it sweeps across the shore and lights up the sand 2 km from the lighthouse is approximately (5 pi / 3) meters/second.