Calculus 1: The illumination of an object by a light source is directly proportional to the strength of the sourcee and inversely proportional to the square of the distance from the source. if 2 light sources, one 3 times as strong as the other, are placed 10 ft apart, where should the object be placed on the line between the sources so as to receive the least illumination
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1. Choose a coordinate system on that line between the two sources.
2. Write down the light intensity as a function in these coordinates. You can take all proportionality constants as 1 because they don't influence extremal points.
3. Find the local and then global extrema of that function.
How do I write down the light intensity as a function?
You have to write the distance in terms of your coordinate and add the intensities of the two lamps together.
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to get you started we know our function needs to be in terms of distance because thats what they want as an answer.
I = illumination
s=strength of lamp 1
d=distance from lamp 1
direct proportion means s goes up, I goes up
inverse proportion means d^2 goes up, I goes down
I = s/d^2
I = 3s/(10-d)^2
second lamp 3X as strong and total space between lamps is 10 so distance from lamp 2 is 10-d
like nowhereman said we have to add them together
I = s/d^2 + 3s/(10-d)^2
Now the goal is to minimize illumination
Differentiate with respect to d and set equal to 0
solve for d