A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks d miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is given by

Question

S = \frac{c}{x^2} + \frac{k}{(d-x)^2}
where c and k are positive constants which depend on the quantity of smoke each stack is emitting. If k = 9 c, find the point on the line joining the stacks where the concentration of the deposit is a minimum.

I know this question has to do with optimizing but I am having a hard time taking the derivative of the equation

Answer 1

You probably expected this to come out as

S = c/x^2 + k/(d-x)^2
= c x^-2 + k(d-x)^-2
dS/dx = -2cx^-3 - 2k(d-x)^-3
= 0 for a min

c/x^3 = k/(d-x)^3
but k = 9c
c/x^3 = 9c/(d-x)^3
1/x^3 = 9/(d-x)^3
(d-x)^3 = 9x^3
d-x = (9^(1/3))x
d = x( 9^(1/3) - 1)

x = d/(9^(1/3) - 1)