calculus
posted by Tori .
Consider the illustration, which shows a rotating beam of light located 0.5 mile from a shoreline. The beam rotates at a rate of 4 revolutions per minute. How fast (in miles per minute) is the distance between the beam and the point where it strikes the shore changing at the instant when x = .25 miles?

Believe it or not, my crystal ball is working to today and I can "see" your illustration.
let the angle formed be Ø, then
dØ/dt = 4(2π) or 8π rad/min
tan Ø = x/(1/2)
tanØ = 2x
sec^2 Ø dØ/dt = 2 dx/dt
dx/dt = (sec^2 Ø)(dØ/dt)/2
when x = 2.5, tan Ø = 2.5/.5 = 5/1
then cosØ = 1/√26
secØ = (3x^4 + 6x^2  48x + 8)26
sec^2 Ø = 26
dx/dt = (26)(8π)/2 = 1004π miles/min 
third last line should have been
sec Ø = √26
(on my Mac, I create √ by pressing "option" V, but I must have pressed "command" V, giving me a "cutand paste" of some previous post) 
tan(Ø)=x/0.5
cos(Ø)=0.5/sqrt(x^2+0.5^5)
sec^2(Ø)=(x^2+0.5^2)
dØ/dt=4 * 2π = 8π rad/min
x=0.5 tan(Ø)
dx/dt=0.5 sec^2(Ø) dØ/dt
dx/dt=(x^2+0.5^2)/0.5 * 8π
when x=0.25 miles
dx/dt=(0.25^2+0.5^2)/0.5 * 8π = 5π = 15.7 miles/min
(note to about attempt: if it were x=2.5, then dx/dt=104π miles/min) 
Oops! Third line should be:
ec^2(Ø)=(x^2+0.5^2)/0.5^2 
Um ...
sec^2(Ø)=(x^2+0.5^2)/0.5^2