Expert: Scotto - 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
1) Lighthouse on island 3 km from point on shore.
Lighthoulse makes 4 revolutions per minutes.
How fast is line moving on shore when it is 1 km from P?

Let S be the distance from the point of light to P on the shore.
Let W be the distance to P over water to the lighthouse.
Let Θ be the angle of the lighthouse.

The two variables of interest are the distance from P and the angle of the lighthouse.
It is known that S/W = tanΘ, where W is a constant and S and Θ are the two variables
about which this problem is concerned.  W is a contant.

This can be rewritten as S(t)/W = tan(Θ(t)), since only S and Θ vary with time.
Taking the derivative, S'(t)/W = sec²(Θ(t))Θ'(t).

W is known, Θ at the point can be determined as tanΘ = 1/3, and Θ' = 1440 rev/min.



lim (x-^2) arctan((x²-4)/(5x²-10x))?
Well, x²-4=(x-2)(x+2) and 5x²-10x=5x(x-2), so cancel (x-2) in the numerator and the denominator.
This gives lim(x->2)Θ(t) where Θ(t) = arctan((x+2)/5).  Put in 2 for x in the equation.
Θ = tan(4/5).  Take the angle in the 1st quadrant.  Add on nπ for any integer n.