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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 three times as strong as the other, are placed 10 ft apart, how far in ft from the stronger source should an object be placed on the line between the sources so as to receive the least TOTAL illumination (the sum of the illumination from each source)?


Let x be the distance from the stronger source, in feet. The distance from the weaker source is therefore (10 - x). Let the total radiant power of the brighter souce be 3P and the other equal to P.

The received illumination is
Y = 3P/x^2 + P/(10-x)^2

Differentiate that to find where dY/dx = 0. Mimimum illumination will be there. You won't need to know P; it will cancel out.

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