The illuminance of a small area changes from 900 lumen m ⁻² to 400 lumen m ⁻² when it is shifted by 10 cm along its normal. Assuming illumination by a point source positioned on the normal, determine the distance between the area and the source in the initial position.

Step-by-step solution

The illuminance E produced by a point source at a distance r is given by E = I r^2 , where I is the luminous intensity of the source. Let the initial distance be r . The initial illuminance is E₁ = 900 lumen m ⁻² . Since the illuminance decreases to E₂ = 400 lumen m ⁻² , the distance must have increased. The new distance is r + 10 cm . Using the formula for illuminance: E₁ = I r^2 = 900 E₂ = I (r + 10)^2 = 400 Dividing the two equations: E₁ E₂ = (r + 10)^2 r^2 = 900 400 ( r + 10 r )^2 = 9 4 Taking the square root o