Question

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

Answer 1

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.

Answer 2

How do I write down the light intensity as a function?

Answer 3

You have to write the distance in terms of your coordinate and add the intensities of the two lamps together.

Answer 4

to get you started we know our function needs to be in terms of distance because thats what they want as an answer.
define variables
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
Lamp 1:
I = s/d^2
Lamp 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
Total Illumination:
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

Answer 5

tag!!