# All Genius' are Welcome!

1. Nov 8, 2009

### zaboda42

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
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

Last edited: Nov 8, 2009
2. Nov 8, 2009

### 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.

3. Nov 8, 2009

### zaboda42

d)

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

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

4. Nov 8, 2009

### 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, whats 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 thats greater than 10?

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

Last edited: Nov 8, 2009
5. Nov 8, 2009

### 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.

6. Nov 8, 2009

Welcome