How Fast Does Light Strength Change with Angle in a Searchlight Problem?

[zaboda42]

Simple yet challenging calc problem - please help!

http://img252.imageshack.us/i/en8t.jpg/

a searchlight is located at point A 40 feet from a wall. The searchlight revolves counterclockwise at a rate of π/30 radians per second. At any point B on the
wall, the strength of the light L, is inversely proportional to the square of the distance d from A; that is, at any point on the wall L = k/d^2 . At the closest point P, L = 10,000 lumens.

a) Find the constant of proportionality k.
b) Express L as a function of θ , the angle formed by AP and AB.
c) How fast (in lumens/second) is the strength of the light changing when θ =π/4? Is it
increasing or decreasing? Justify your answer.
d) Find the value of θ between θ =0 and θ =π/2 after which L<1000 lumens.

I was only able to do part a) the other parts are rather confusing. If anyone could help that would be greatly appreciated.
My Attempt:

a) (40)^2 + (10,000)^2 = d^2
d = 10,000

L = k/d^2
10,000 = k/(10,000)^2
k = 1E12

 

[zaboda42]

b)

L = [.5(40 x 10,000)] x sin(theta)
L = 200,000 x sin(theta)

I'm sorry, I'm really confused and just need guidance to know if I'm doing this problem right, or if I'm completely wrong.

 

[zaboda42]

d)

999.999 =< 200,000 x sin(theta)
theta =< 0.005

I have a feeling I'm doing this terribly wrong =/

 

[ƒ(x)]

For A, you are given three unknowns (one of which is a constant) and the values for the two variables (what is the distance between point A and point P?). I'm sure that you can figure A out from there.

For B, what's the relationship between d and theta?

For C, think instantaneous rate of change.

For D, how what would you do if you were asked to find of x for y = 2x so that every value of x after that produced a y value that's greater than 10?

My answers are below in a spoiler so that you can check yours.

Spoiler

a) L(d) = k/d²
L(40) = 10000
10000 = k/40²
16000000 (1.6 x 10⁷) = k

b) cos(θ) = A/H = 40/d
d = 40/cos(θ)

L(θ) = k/(40/cos(θ))² = 10000cos²(θ)

c) L(θ) = 10000cos²(θ)
dL/dθ = 10000•2•cos(θ)•-sin(θ) = -20000cos(θ)sin(θ)
when θ = π/4 --> dL/dθ = -20000cos(π/4)sin(π/4) = -10000
Since the derivative is negative, L is decreasing when θ = π/4

d) L(θ) = 1000
1000 = 10000cos²(θ)
1/10 = cos²(θ)
1/\sqrt{10} = cos(θ)
cos^-1(1/\sqrt{10}) = θ
which is about .398π

 

[zaboda42]

Thanks so much ƒ(x), i personally loved your "hints" in the beginning because they really helped me look at the problems differently rather than me just getting an answer.

Thanks again.

 

[ƒ(x)]

Welcome