# Finding the darkest point between two lamps

I was working on some exercises when I came over a rather curious question.

Two lamps have intensities 40 and 5 candle-power and are 6 m apart. If the intensity of illumination I at any point is directly proportional to the power of the source, and inversely proportional to the square of the distance from the source, find the darkest point on the line joining the two lamps.

What puzzled me was how the intensity at any given point is both directly proportional to the power of the source, and inversely proportional to the square of the distance from the source.

Assuming only the latter statement I thought maybe I could figure it out by the equation $I=\frac{1}{x^2}+\frac{1}{(6-x)^2}$ and then finding when $\frac{dI}{dx}=0$. However that gave me 3, which is just the midpoint between the two points and does not rely at all on the brightness of each lamp. The book gives the answer 4m from the brightest lamp, but I am not certain on how to get there.

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The equation you wrote doesn't factor in the strength of the light. Try the equation $$I = \frac{40}{x^2} + \frac{5}{(6-x)^2}$$