
Problem Help Needed
After a rowdy party, a group of intoxicated youth stumbles into the county fairground. One of these youngsters, Peter, climbs to the middle of a spinaround carousel, and his friends push the carousel so that it rotates once every ten seconds. Peter, who is trying not to get sick, holds a flashlight motionless in his hand.
There is a road running by the edge of the fairground that, at its closest point, is 30 yards from the middle of the carousel. Unknown to Peter, there are two security offices watching him from the road, amused at the spectacle and Peter's increasingly colorless complexion. One of them is standing on the road at the point closest to the carousel, while the other is standing 45 yards down the road. At approximately what speed (in yards per second) does the spot illuminated by the flashlight traverse each of the security officers' bodies?
State speed at the closest officer first.

What ideas have you had so far?

well it is interesting. I preface by stating that my ideas may not be sound in judgment in regards to the actual answer.
So i started by drawing a circle to represent the carousel. I put a point in the middle to represent peter. from that point, i drew a line downwards and let it intersect a new line (which represents the road). I marked that distance as 30 yards between the two points, and as the shortest distance between two points is a straight line, i made the intersection of the lines perpendicular.
second, on the line that represents the road, i made another point towards the left, but still on the line. i marked this with a distance of 45 yards. so at this point i have a triangle with the length of the legs being 30 and 45. as i made it a right triangle, i used the pythagorean theorem to find the length of the hypotenuse (15 root13). that is about as far as i have got, and the sad part is that my method may be completely unneeded.
any ideas on solving it, or if my methods are correct in any ways?

You're on the right track. That is, none of the computations you've done so far are at all superfluous. The question is asking for the tangential component of the velocity of the light beam at two different radii. How would you compute that?

breakthrough!!!!! or so i think.
So building off the previous foundation. using my trusty calculator and the inverse tangent function, i found the angle of the two intersecting lines at the carosel to be around 56.39 degrees. with that i went to this next step.
With the line that is marked at length 30 (at where officer one is standing), i drew a massive circle with 30 being the radius. i found the circumference then to be 188.5 yards. as we already know that it takes 10 seconds for a full rotation, i simply divided 188.5 by 10 to give me 18.85 yards per second.
I repeated this step for the other officer using the length 15 root 13 as my radius. after computing, i found an answer of roughly 33.98 yards per second. I feel as though my explanation here is vague (I have massive troubles explaining how i do things, especically when i do so in writing). but i do hope you were able to follow. I do want to ask though, if what i have done is correct or near the correct path. thanks so much for your feedback and willingness to listen through my awful typing skills.

I think you're done. I don't think your explanation is all that poor: it's clear enough to me what you've done. Good job!