# Trouble with rates of change question

A lamp on a lighthouse rotates exactly three
times every minute. The lighthouse is
80 metres from a straight shoreline.

a. Calculate how fast the beam from the
lighthouse is moving along the shoreline
at a point directly opposite the lighthouse.
b. A couple note that the beam travels
past their location at a speed of 50 m/s.
Calculate the couple’s distance from the
lighthouse, to the nearest metre.

I managed to get a but I’m stuck on b as two lengths of the triangle are unknown?

To answer this question we, in fact, don’t need to use differentiation. The speed of a beam as it moves past a particular point on circle can be calculated as v =D/t, where D - circumference of the circle and t - time it takes to make full rotation.

• D = 2Pir = 2Pi*80 = 160Pi (m)
t = 20s as the beam makes 3 revolutions per minute, it will take 20sec to make one full rotation.
Therefore v = 160Pi/20 = 8Pi = 25.1 (m/s)

• A couple is somewhere around the lighthouse, we don’t know if they are on the shore or not, we only need to find their distance from the lighthouse, or radius of the circle.
D = v*t = 50 * 20 = 1000m that is circumference of this circle.
r = 1000/2Pi = 159m