Question

A searchlight rotates at a rate of 2 revolutions per minute. The beam hits a wall located 8 miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per hour) is this dot moving when the angle between the beam and the line through the searchlight perpendicular to the wall is pi/5? Note that d angle,dt=2(2pi)=4pi.

Speed of dot _______ = mph.

Answer 1

draw the triangle
s=Searchlight postion
8= distance wall is away.
x= distance from the perpendicular to the light to the position of the beam, so that

TanTheta=x/8

take derivative w/respect to time
d(tanTheta)=1/8 dx/dt
sec^2 theta * dtheta/dt=1/8 dx/dt

you are given theta, given dtheta/dt (4PI), find dx/dt

Answer 2

dθ/dt=2 rev/min=4π rad/min

dx/dt=8*(sec^2(π/5))*8π=154 miles/min=9216 mph