# Calculus: Modelling and Optimization

Hi guys, me and my fellow classmate have been working on a math problem we believe to be a Modeling and Optimization type problem in our calculus class. We've been at it for 2 days now! Just can't seem to figure it out... We'd really like appreciate and all help!

## Homework Statement

[There is a picture shown of a right triangle. "d" is the hypotenuse that connects points A to B. Lines BP and AP connect to form a right angle at point P. Line AP measures 40 feet. Line BP is labelled as "Wall"]

A searchlight is located at point A, 40 feet from a wall. The searchlight revolves counterclockwise at a rate of ∏/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/d2. 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=∏/4? Is it increasing or decreasing? Justify your answer.

D) Find the value of θ between θ=0 and θ=∏/2 after which L < 1000 lumens.

L = k/d2

## The Attempt at a Solution

A) We rearranged the equation given and got Ld2=k

B) Used cosine of the angle A:

cosθ=40/d
d=40/cosθ

Then plugged this into the original equation (L=k/d2):

L=k/(40/cosθ)2

C) This is where we got stuck.. The farthest we came up with is this:

We are given: dθ/dt=∏/30
We have to find: dL/dt
When: θ=∏/4

But the equation we created in B does not suffice for what we have to find.
Or maybe it does... We're not sure, could someone bring light to this?

D) No clue. We just know that we're looking for a value in the range [0,∏/2], so it will be located within the first quadrant of the graph.

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