# Thread: Proportions & Integrals

1. ## Proportions & Integrals

A searchlight is located at point A, 40 feet from a wall. The searchlight revolves counterclockwise at a rate of pi/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 frm 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 thetak the angle formed by AP and AB
C) how fast(in lumens per second) is the strength of the light changing when theta=pi/4? Is it increasing or decreasing? Justify your answer.
D) Find the value of theta between theta=0 and theta=pi/2 after which L is less than 1000 lumens

2. Originally Posted by rawkstar
A searchlight is located at point A, 40 feet from a wall. The searchlight revolves counterclockwise at a rate of pi/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 frm 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.

$\textcolor{red}{10000 = \frac{k}{40^2}}$ ... solve for k

B) express L as a function of thetak the angle formed by AP and AB

$\textcolor{red}{\cos{\theta} = \frac{40}{d}}$

solve for d in terms of $\textcolor{red}{\theta}$, then determine L as a function of $\textcolor{red}{\theta}$

C) how fast(in lumens per second) is the strength of the light changing when theta=pi/4? Is it increasing or decreasing? Justify your answer.

determine the value of $\textcolor{red}{\frac{dL}{dt}}$ at the stated angle.

D) Find the value of theta between theta=0 and theta=pi/2 after which L is less than 1000 lumens

set the expression for L < 1000 and solve for $\textcolor{red}{\theta}$
...

3. Originally Posted by rawkstar
A searchlight is located at point A, 40 feet from a wall. The searchlight revolves counterclockwise at a rate of pi/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 frm 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 thetak the angle formed by AP and AB
C) how fast(in lumens per second) is the strength of the light changing when theta=pi/4? Is it increasing or decreasing? Justify your answer.
D) Find the value of theta between theta=0 and theta=pi/2 after which L is less than 1000 lumens
A) Find the constant of proportionality k.

At the closest point P, L=10,000 Lumens
A searchlight is located at point A, 40 feet from a wall. So d = 40
Use L=k/d^2 to solve for k

B) express L as a function of the angle formed by AP and AB

I you connect A, P and B you will have a right triangle with AP=40, AB=d and measure of angle A= $\theta$

4. Originally Posted by skeeter
...

im not sure how you would solve cos theta=40/d other than d=40/cos theta
and then i have no idea how to determine L as a function of theta

5. Originally Posted by rawkstar
im not sure how you would solve cos theta=40/d other than d=40/cos theta
and then i have no idea how to determine L as a function of theta
you're halfway there ... substitute $\frac{40}{\cos{\theta}}$ in for $d$ in the equation for $L$ and simplify as much as possible.