math question ~~~~~~~~~~~~~~~~~~~~~~~~?

Question

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one four times as strong as the other, are placed 18 m apart, how far away from the stronger light source should an object be placed on the line between the two sources so as to receive the least illumination?

plz help

Answer 1

13.74 m

Answer 2

Let I1 and I2 be the illumination levels from the two sources, with Si and Di being the corresponding strengths and distances. Then:

I1 = S1/(D1^2)
I2 = S2/(D2^2)

You know S1 = 4 S2 and D1 + D2 = 18

Now you want to find the D2 that minimizes I1 + I2

Recast I1 + I2 as a function of S2 and D2 by substituting for S1 and D1.

Then differentiate with respect to D2. The minimum value will be either at the end points, at a point where the derivative is 0, or at a point where the derivative does not exist.

In this case it is clear that it is not at the endpoints and that the derivative exists in between them, so you want a point where the derivative is 0.