A rotating beacon is located 7 miles out in the water. Let A be the point on the shore that is closest to the beacon. As the beacon rotates at 8 rev/min, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. How fast is the point where the beam hits the shore moving along the shore at an instant when the beam is lighting up a point 7 miles along the shore from the point A?

Question

anonymous · Answer

I think it's going to be 7 * 8 = A

anonymous · Answer

I was thinking that it would be a triangle, and maybe applied derivatives of pythagarym theory?

anonymous · Answer

I think so too.......cuz I'm not good at problem solving but u could give me a medal for trying to answer ur question if u want

turingtest · Answer

@kludden you are very much on the right track

anonymous · Answer

Okay, I also looked at taking the derivative of tan(theta)= x/7   
I'm not sure how to find out dx/dt though.

turingtest · Answer

dx/dt is what you are solving for
what did you get after differentiating with respect to time the expression you wrote?

anonymous · Answer

for the tan equation or for the pythagaram theory?

anonymous · Answer

dx/dt should be the given rate that the light is travelling right?

turingtest · Answer

ye
first of all, do you know d(theta)/dt  ?

anonymous · Answer

no, I was thinking that would be what we were trying to find

turingtest · Answer

you can figure out d(theta)/dt from the fact that the thing is revolving at 8rev/min
how many radians in one rev?

anonymous · Answer

If the beam moves in a circle 7 miles from shore, then youre looking at a circumfrence of 2pi*7. so 14pi. It moves 8 rev per minute, so 112pi ft/min.

anonymous · Answer

that could be incorrect

anonymous · Answer

2pi radians in a circle, so 10pi in 1 minute

turingtest · Answer

1revolutions is 2pi radians, so 8 rev/min=16pi rad/min=d(theta)/dt

turingtest · Answer

now differentiate$$\tan\theta=\frac x7$$with respect to time, what do you get?

anonymous · Answer

wow, yes that. My multiplication skills are superb lol. so 16pi rad in one minute. that gives you d(theta)/dt.  You would get (1/7)*(1/(1+((x/7)^2))

anonymous · Answer

is equal to d(theta)/dt

turingtest · Answer

whoa, you differentiated wrong
what is$$\frac d{dt}\tan\theta$$?

anonymous · Answer

sec^2
okay I took arctan to isolate theta
but we have that so I was doing it wrong.

you would get d(theta)/dt sec^2(theta)= (1/7) dx/dt

turingtest · Answer

yes, much better
now just solve for dx/dt :)

anonymous · Answer

okay! Thank you so much!

turingtest · Answer

welcome!