## anonymous 5 years ago A revolving search light, which is 800 yards from the shore, makes 2 revolutions (4 radians) per minute. How fast is the light traveling along the straight beach when it is 1000 yards from the lighthouse? HELP

1. anonymous

is this a calculus problem?

2. anonymous

yea

3. anonymous

what topic is it in ?

4. anonymous

thats what i was thinking .....because its right off the final review

5. anonymous

damn you are typing a lot rob O.o

6. anonymous

is this calc 1 mmonish91 ?

7. anonymous

yes

8. anonymous

sorry... i can't think of the topic this question involves

9. anonymous

10. anonymous

Imagine a rigid rod 1000 yards long, attached to the searchlight rotating @ 2 revolutions per minute or 120 revolutions per hour. The circumference of a circle of radius 1000 is 2000 Pi. Multiply 2000 Pi by 120 revolutions per hour and by 3 to convert to feet. 2000 * Pi * 120 * 3 = 720000 Pi feet /hr The tip of the rod travels (720000 Pi)/5280 or 428.4 miles per hour over the beach in a circular motion. I hope I got this right.

11. anonymous

hmm.. interesting

12. watchmath

$$\cos \theta = y/800$$. We want to find $$dy/dt$$ when $$y=1000$$. Take the derivative implicitly we have $$-\sin\theta \cdot d\theta/dt=1/800\cdot dy/dt\qquad(*)$$ When $$y=1000$$ the opposite side of the angle theta is $$\sqrt{1000^2-800^2}=360$$. In that case $$\sin\theta = 360/1000=3.6$$ plug in $$\sin \theta =3.5$$ and $$y=1000$$ to $$(*)$$ we have $$-3.6\cdot 4\pi=1/800\cdot dy/dt$$ $$dy/dt=-3.6\cdot 4\pi\cdot 800$$ yards/minute.

13. watchmath

Sorry the length of the opposite side should be $$600$$ and $$\sin \theta =0.6$$ So the answer is $$dy/dt=-0.6\cdot 4\pi\cdot 800$$

14. anonymous

@watchmath-\[\cos 4\pi = 800/y\cosine is adjacent over hyp

15. watchmath

That is correct. The y is the hypothenuse, the 800 is the adjacent.

16. anonymous

yes, so the one that i gave is correct?

17. watchmath

The theta is also changing with respect to t. So we can only say that $$\cos(\theta)=y/800$$

18. anonymous

but, we are talking here with the cosine.

19. watchmath

what are you trying to say about cosine function?

20. anonymous

21. watchmath

The $$4\pi$$ is the rate of how the theta changes, i.e., $$d\theta/dt=4\pi$$ and not an actual angle itself.

22. watchmath

I agree with you, what I didn't agree is that you plug in $$4\pi$$ for the angle.

23. anonymous

yeah..i don't have problem with that.

24. anonymous

i'm sorry, it must be no 4.

25. watchmath

Ah I see... what you meant now ....yes. I should write 800/y instead of y/800. I didn't draw the picture. I just did it in my head :D. Thanks for the correction.

26. anonymous

lol, you are welcome:)

27. watchmath

(Please check again pat18) I think I have to fix this before I off go to bed $$\cos \theta =800/y$$ $$-\sin\theta\cdot d\theta/dt=-800/y^2\cdot dy/dt$$ $$0.6\cdot 4\pi=800/10^6\cdot dy/dt$$ $$dy/dt=3000\pi\text{ yards/minute}$$ (I am not good at physics, is that make sense?)

28. anonymous

you are welcome.