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**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.

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

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

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

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

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 $\displaystyle \textcolor{red}{\frac{dL}{dt}}$ 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 $\displaystyle \textcolor{red}{\theta}$