Calculus
posted by Anonymous on .
A lighthouse is located on a small island 4 km away from the nearest
point P on a straight shoreline. Its light makes 3 revolutions
per minute. How fast is the beam of light moving along the shoreline
when it is 1.7 kilometers from
P
?

Draw a diagram. If x is the distance from P, and θ is the angle swept out by the beam, measured from the line joining the lighthouse and P,
x/4 = tan θ
dx/4 = sec^2 θ dθ
dx = 4 sec^2 θ * 3 * 2π
So, what is sec θ? tan θ = 1.7/4 = 0.425
sec^2θ = 1+tan^2θ = 1.18
dx = 4*1.18*3*2π = 88.97 mi/min