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 d miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is given by
S = cx^-2+k(d-x)^2 k=9c find minimum
need help with back half of question, please

Answer 1

What is c?

Answer 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.

Answer 3

So I get S= cx^-2+9c(d-x)^-2 and factor out the c for later to get x^-2+9(d-x)^-2

Answer 4

Well, first you need to take the derivative with respect to x. Once you have that function, set it equal to 0 and solve for x.

Answer 5

yup. I get x=d/3.080083823, right?

Answer 6

Now what?

Answer 7

Where did c go?

Answer 8

so, cd/3.080083823

Answer 9

all these variables really screw me up

Answer 10

ok, what do you get as the derivative of S(x)

Answer 11

-2x^-3+18(d-x)^-3

Answer 12

ok you're close. For the power rule of differentiation, you multiply by the power, then lower the exponent.

Answer 13

where didnt I do that? I even applied the chain rule for the second

Answer 14

oops, I read it wrong. Sorry about that. What you have is right.

Answer 15

so what do u get for x? and what do I do with that info?

Answer 16

So setting that equal to 0, you get 2c/(x^3) = 18c/(d-x)^3
1/x^3 = 9/(d-x)^3
(d-x)^3/x^3 = 9
Solve that for x and your answer should only be in terms of d.
Sorry. I think I should be the one asking the question =p

Answer 17

Take the x value you get that is in terms of d, and plug it back into the original equation to find the minimum value for S(x).

Answer 18

Well the deadline was midnight and I had it right the whole time just entered it wrong.. :(

Answer 19

d/(1+2.08)

Answer 20

thanks for the feedback, seriously!

Answer 21

Haha, glad I could "help".