# Calculus

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Consider the illustration, which shows a rotating beam of light located 0.5 mile from a shoreline. The beam rotates at a rate of 4 revolutions per minute. How fast (in miles per minute) is the distance between the beam and the point where it strikes the shore changing at the instant when x = .25 miles?

• Calculus -

It depends upon the angle that the shore makes with the beam.

What is x ?

An illustration is needed for me to make sense of this question.

• Calculus -

I recall seeing this type of question before, and I will assume it is the standard type.
Usually we are to find the speed of the light as it moves along the shore.

Draw a perpendicular line from the lighthouse to the shore.
Let the angle between the line to the shore and the beam of light be Ø , and let the light rotate counterclockwise.
Let the beam of light be x miles along the shore
tanØ = x/.5
x = .5tanØ
dx/dt = .5 sec^2 Ø dØ/dt
when x = .25
tanØ = .25/.5 = 1/2
cosØ = 2/√5
secØ = √5/2
sec^2 Ø = 5/4

dx = .5(5/4)(8π) miles/min = 5π miles/minute

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