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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, where should an object be placed on the line between the sources so as to receive the least illumination?

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Irving Heathcote
Irving HeathcoteLv2
23 May 2020
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Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on, so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.

(a) Find an expression for the intensity at the point P.

(b) If m, use graphs of and to show that the intensity is minimized when m, that is, when P is at the midpoint of ,.

(c) If m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint.

(d) Somewhere between m and m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d.

A light is to be placed atop a pole of height h feet to illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination I at any point P on the circle

is directly proportional to the cosine of the angle ‑ (seethe figure) and inversely proportional to the square of the distance d from the source.

(a) How tall should the light pole be to maximize I?

(b) Suppose that the light pole is h feet tall and that a woman is walking away from the base of the pole at the rate of 4 ftys. At what rate is the intensity of the light at the point on her back 4 ft above the ground decreasing when she reaches the outer edge of the

traffic circle?

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