Expert: Paul Klarreich - 2/2/2012

Question
A lamp is hung above a circular table of radius r=15in at a variable height h. As h varies, the distance l from the lamp to the edge of the table and the angle (p) also vary. Use cos to eliminate l from the formula I=k((sinp)/(l^2). You will introduce r, but this is a constant. Instead of maximizing I, try to maximize F=((r^4)/(4k^2))I^2. Hint you will need to use Pythagoras to make a substitution for sin^2p, and also use the fact that 1/4=(1/2)(1/2).

I started on the first part of the problem in trying to eliminate the l. So I changed I=k((sinp)/(l^2)) into (cosp)/2=(1/l) then I changed that to cosp=(2/l). That is were I got stuck and I don't know if I even did that first part right.

Answer
Questioner:Alexis
Country:Wisconsin, United States
Category:Calculus
Private:No
Subject:max/min question

Question:

A lamp is hung above a circular table of radius r=15in at a variable height h. As h varies, the distance l from the lamp to the edge of the table and the angle (p) also vary. Use cos to eliminate l from the formula I=k((sinp)/(l^2). You will introduce r, but this is a constant. Instead of maximizing I, try to maximize F=((r^4)/(4k^2))I^2. Hint you will need to use Pythagoras to make a substitution for sin^2p, and also use the fact that 1/4=(1/2)(1/2).

I started on the first part of the problem in trying to eliminate the l. So I changed I=k((sinp)/(l^2)) into (cosp)/2=(1/l) then I changed that to cosp=(2/l). That is were I got stuck and I don't know if I even did that first part right.
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You have not at all made clear how this is a max/min problem.  I think if you start with that, it may be clear what to do.

It looks to me as if you want to maximize the light intensity at the edge of the table.

Is that it?

If so, write an expression for I in terms of h and go on from there.

Did you look at my list of max-min problems?