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A differential problemby Bob777
Tags: differential 
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#1
Mar1512, 10:57 PM

P: 4

1. The problem statement, all variables and given/known data
A light is to be placed atop a pole of height h feet to illuminate a busy traffic circle, which has a radius of 40ft. The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle θ (see the figure) and inversely proportional to the square of the distance d from the source. (a)How tall should the light pole be to maximize I? 2. Relevant equations 3. The attempt at a solution intensity of illumination I=cosθ/d^2 cosθ/d^2*the area=I but any point P has different intensity of illumination according to the distance form center. and when h change all the intensity of illumination of any point change. How can I know the I? Please help me! thanks 


#2
Mar1612, 01:00 PM

P: 312

write d as a function of θ, which seems straightforward, then I=cosθ/d(θ)^2, now you can take the derivative



#3
Mar1712, 07:24 AM

P: 4

It's "intensity of I" "intensity of I"=cosθ/d(θ)^2 points have different θ with different radius. and when h changes everything changes. :( 


#4
Mar1712, 08:08 PM

P: 312

A differential problem
The intensity you want to maximize is cosθ/d(θ)^2, which is a function of θ, you want to choose a θ, so that this intensity is maximized. How do you choose θ? You realize that when the intensity is maximized, its derivative with respect to θ is zero... now try to work out the rest. Regarding h, think of h changes with θ, not the contrary



#5
Mar1912, 11:37 PM

P: 4

My question is that the intensity is different at different points on the circle. I don't know which point on circle it's intensity I have to find the h to maximize . Should I find maxim intensity of all points on the circle? 


#6
Mar2012, 07:56 AM

Math
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Thanks
PF Gold
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#7
Mar2312, 03:49 AM

P: 4

Now there is no problem. :) 


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