If the telescope is 100 feet above the water level and if the boat is approaching at 10 feet per seconds, at what rate is the angle of the telescope changing when the boat is 150 feet from the shore?

Question

A woman standing on a cliff is watching a motorboat through a telescope as the boat approaches the shoreline directly below her. If the telescope is 100 feet above the water level and if the boat is approaching at 10 feet per seconds, at what rate is the angle of the telescope changing when the boat is 150 feet from the shore?

negative ______ radian per second

Answer 1

L = 10cotθ; dL/dt = - 10csc²θdθ/dt; dθ/dt = - 0.1dL/dtsin²θ; dL/dt = ·0.1·10· 100/(150² - 100) = 0.0044643 rad/s

Answer 2

y=150 condition depends on dy/dt

dy/dt=-10

x=100 is constant

tanθ(t)=y(t)/x

d(tanθ)/dt=(1/100)*dy/dt

sec²θ*(dθ/dt)=(1/100)*(-10) ,,,dy/dt=-10

dθ/dt=[(10⁴/(10⁴+2.25*10⁴)](1/100)*(-10) ,,,,at y=150 and x=100

dθ/dt=-(1/32.5) radian/second