# calculus

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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 11 ft apart, how far from the stronger source should an object be placed on the line between the sources so as to receive the least illumination? Round the result to the nearest hundredth.

a. 6.57 ft

b. 6.5 ft

c. 4.8 ft

d. 6.7 ft

e. 6.55 ft

f. 5.4 ft

• calculus -

let x ft be the distance from the stronger light, so 11-x is the distance from the weaker light

let TI be the total illumination, then

TL = 3I/x^2 + I/(11-x)^2
d(TL)/dx = -6I(x)^-3 - I(11-x)^-3(-1)
= 0 for a min of TL

divide both sides by I, cross-multiplying and reducing I go
x^3 = 3(11-x)^3

take the cube root of both sides:
x = 3^(1/3)(11-x)

expand the right side, bring the x term to the left, take out a common factor of x, ...
I got

x = 11(3)^(1/3) / (1+(3)^(1/3))
= 6.4959..

looks like b)

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