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:
[tex]S=\frac{c}{x^2} +\frac{k}{(d-x)^2}[/tex]
where c and k are positive constants which depend on the quantity of smoke each stack is emitting. If k=8c, find the point on the line joining the stacks where the concentration of the deposit is a minimum. xmin= ? miles

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sqdancefan sqdancefan · Answer 1 · 2021-11-23T04:01:01+01:00

Answer:

d/3

Step-by-step explanation:

For k = 8c, the concentration is ...

S = c/x^2 + 8c/(d -x)^2

The minimum of the function will be found where its derivative is zero. The derivative with respect to x is ...

S' = -2c/x^3 +(-2)(8c)(-x)/(d -x)^3

We are interested in the value of x that makes this zero.

0 = -2c/x^3 +16c/(d -x)^3 . . . . . simplify slightly, set S'=0

0 = (-2(d -x)^3 +16x^3)/(x(d -x))^3 . . . . . combine terms, divide by c

0 = d^3 -3d^2x +3dx^2 -x^3 -8x^3 . . . . multiply by (x(d-x))^3/(-2)

9x^3 -3dx^2 +3d^2x -d^3 = 0 . . . . . simplified cubic

Factoring by grouping, we have ...

(9x^3 -3dx^2) +(3d^2x -d^3) = 0

3x^2(3x -d) +d^2(3x -d) = 0

(3x^2 +d^2)(3x -d) = 0

The solution will be found where the second of these factors is zero:

3x -d = 0

x = d/3

The concentration of the deposit is a minimum at x = d/3 miles.

_____

The graph shows the minimum is x = 1/3 when d = 1.