Expert: Sherry Wallin - 11/22/2009

Question
Hey I can't figure these out, do we have enough info?

1)A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? _________km/min

Also I keep getting 21.801 which is wrong for

2)limit as x>2 arctan(x^2-4/5x^2-10x)  

Answer
For #1- the light is moving 4rev/min (constantly). Think of the lighthouse as being the center of a circle and radius as being the distance from the lighthouse to point P. So the distance traveled in one revolution is just the circumference of the circle the light sweeps. C = 2*pi*r = 2*pi(3) = 6*pi km is the distance traveled in 1 revolution, so now you have the following unit change: You want to take information about rev per min and write it as distance per min and you already know the rev/min and now you have 6*pi/rev so 4rev/1min x (6*pi)km/1 rev = (24*pi)km/min. The speed of the light is constant, it doesn't matter where it is being measured so the fact that it asks for the speed 1 km from point P is 'extra', i.e., unnecessary information.

For #2- I am assuming your function is (x^2-4)/(5x^2-10x). The way you have it written it is not that. the way you have it written is equivalent to x^2-(4/x^2)-10x which I don't think you mean so assuimg it is as I first stated factor the argument for arctan ass (x+2)(x-2)/5x(x-2), simplify and you get
lim x-> 2 arctan[(x+2)/5x] = arctan(4/10 = 2/5) which is 21.801 degrees. Why do you think yours is wrong? Maybe the answer you are comparing this to is in radians? You want the angles whose tangent is .4

Math Prof