A searchlight rotates at a rate of 3 revolutions per minute. The beam hits a wall located 7 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 heta between the beam and the line through the searchlight perpendicular to the wall is \frac{\pi}{6}? Note that d heta/dt=3(2\pi)=6\pi.

Question

A searchlight rotates at a rate of 3 revolutions per minute. The beam hits a wall located 7 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 	heta between the beam and the line through the searchlight perpendicular to the wall is \frac{\pi}{6}? Note that d	heta/dt=3(2\pi)=6\pi.

Nick Kramer · Accepted Answer

Thank you this helped me

Damon · Answer

3 * 2 pi = 6 pi radians/minute

pi/6 = 30 degrees by the way

I call your angle theta A
dA/dt = 6 pi rad/min

tan A = x/7
x = 7 tan A
dx/ dt = 7 d/dt(tan A ) = (7/cos^2A) dA/dt
cos^2 (30) = .75
so
dx/dt = (7 miles/.75)(6 pi rad/min)
dx/dt = 176 miles/min
* 60 = 1055 miles/hr
so
dx/dt =

H H Chau · Answer

Concur. dx/dt=7*(4/3)*2π=176 miles/min=10555miles/hr