Question

The illumination at a point is inversely proportional to the square of the distance of the point from the light source and directly proportional to the intensity of the light source. Suppose two light sources are ð  feet apart and their intensities are ð¼ and ð½, respectively. Let P be the point between them where the sum of their illuminations is a minimum. The distance of P from light source ð¼ will be

Answer 1

Answer:

ð/[2^(1/3)] + 1 or ð/2.26

Step-by-step explanation:

Step 1: Let the illumination be denoted as C

Let the two intensities be denoted as M and N respectively

Let the distance from point P to M be x and distance from P to N be (ð-x

Step 2:

Illumination at point P from M: Cm= kM/x^2

Illumination at point P from N: Cn= kN/(ð-x)^2

The sun of the illumination: Q=Cm+Cn = (kM/x^2) + (kN/(ð-x)^2)

Differentiate Q wrt x and equate to zero, we have

(-2kM/x^3) + [2kN/(ð-x)^3] = 0

Simplifying the above equation, we have

(ð-x)^3/x^3 = -2kN/-2kM

[(ð-x)/x]^3 = N/M

[(ð-x)/x] = (N/M)^(1/3)

ð-x = x[(N/M)^(1/3)]

ð = x[((N/M)^(1/3)) + 1]

Therefore, x = ð/[((N/M)^(1/3)) + 1]

Substitute M= ð/4 and N=ð/2 into equation above, we have

x = ð/[2^(1/3)] +1 or x = ð/2.26