Question

A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks 20 miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is given by
S = ksub1 over x^2 + ksub2/((20-x)^2)
where ksub1 and ksub 2
are positive constants which depend on the quantity of smoke each stack is emitting. If
ksub1 = 2ksub2,
find the point on the line joining the stacks where the concentration of the deposit is a minimum. (Round your answer to two decimal places.)

so:
s= 11k2/x^2 + k2/((20-x)^2) = 11k2((20-x)^2) + x^2k2 all divided by x^2((20-x)^2)
so then I multiplied it out:

11k2(x^2-40x+400)+x^2k2
11k2x^2-440k2x+4400k2+x^2k2

and then I didn't know where to go from there unfortunately

Answer 1

Let's call the k's a and b just to make things more readable.

S = a/x^2 + b/(20-x)^2
If a = 2b, then
S = 2b/x^2 + b/(20-x)^2
Now, we can actually ignore b, since it is just a constant multiplier and will not affect where the minimum occurs. So, let's use

S = 2/x^2 + 1/(20-x)^2
Now we have
dS/sx = -4/x^3 + 2/(20-x)^3
=

-2(3x^3-120x^2+2400x-16000)
--------------------------------
x^3 (20-x)^3

All we need to examine is the numerator.

dS/dx=0 at x ≈ 11.15

Answer 2

Thank you so much for helping me understand how to solve the problem!