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 = \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

To find the point on the line joining the smokestacks where the concentration of the deposit is at a minimum, we need to optimize the function given in the question, which is:

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

where k = 9c.

To find the minimum, we need to take the derivative of S with respect to x and set it equal to zero. Let's go through the steps:

1. Take the derivative of S with respect to x:
dS/dx = -2c/x^3 + 2k/(d-x)^3

2. Set the derivative equal to zero and solve for x:
-2c/x^3 + 2k/(d-x)^3 = 0

Since k = 9c, we can substitute k with 9c:

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

Simplifying:

-2c/x^3 + 18c/(d-x)^3 = 0

3. To simplify the equation further, we can multiply both sides by x^3(d-x)^3 to eliminate the denominators:
-2c(d-x)^3 + 18cx^3 = 0

4. Expanding and rearranging the equation:
-2c(d^3 - 3d^2x + 3dx^2 - x^3) + 18cx^3 = 0
-2c(d^3 - 3d^2x + 3dx^2 - x^3) + 18cx^3 = 0

5. Simplifying:
2c(d^3 - 3d^2x + 3dx^2 - x^3) = 18cx^3
(d^3 - 3d^2x + 3dx^2 - x^3) = 9x^3
d^3 - 3d^2x + 3dx^2 - x^3 = 9x^3
d^3 - 3d^2x + 3dx^2 = 10x^3

6. Rearranging the equation:
10x^3 - 3dx^2 - 3d^2x + d^3 = 0

7. Factoring out an x:
x(10x^2 - 3dx - 3d^2) + d^3 = 0

8. Since we're solving for x, we can ignore the last term:
10x^2 - 3dx - 3d^2 = 0

9. Dividing the equation by d:
10(x^2 - 3x - 3d) = 0

10. Solving the quadratic equation with the quadratic formula or factoring, we find the solutions for x:
x = (3 ± √(9 + 120d)) / 20

Now you can substitute the value of d to find the specific points on the line joining the stacks where the concentration of the deposit is at a minimum.

To find the point on the line joining the smokestacks where the concentration of the deposit is a minimum, we can use calculus and find the minimum point of the concentration function. Let's go step by step to solve this problem.

We have the concentration function:
S = c/x^2 + k/(d-x)^2

Given that k = 9c, we can substitute this value into the concentration function:
S = c/x^2 + 9c/(d-x)^2

Now, to find the minimum point, we need to find the value of x that minimizes the concentration function. To do this, we can take the derivative of the concentration function with respect to x:

dS/dx = -2c/x^3 + 18c/(d-x)^3

To find the critical points, we set the derivative equal to zero:

0 = -2c/x^3 + 18c/(d-x)^3

Next, we can simplify this equation by multiplying through by x^3 and (d-x)^3 to get rid of the denominators:

0 = -2c(d-x)^3 + 18cx^3

Now, let's expand and simplify this equation:

0 = -2c(d^3 - 3d^2x + 3dx^2 - x^3) + 18cx^3

0 = -2cd^3 + 6cd^2x - 6cdx^2 + 2cx^3 + 18cx^3

0 = 20cx^3 - 6cdx^2 + 6cd^2x - 2cd^3

Now, we can factor out the common factor of 2c:

0 = 2c(10x^3 - 3dx^2 + 3d^2x - d^3)

Setting each factor equal to zero, we get two possibilities:

10x^3 - 3dx^2 + 3d^2x - d^3 = 0 (Equation 1)

Or,

2c = 0 (Equation 2)

Since c is a positive constant, Equation 2 implies no solution for this problem.

So, we need to solve Equation 1 to find the value of x.

Unfortunately, there is no general formula for solving cubic equations like Equation 1. However, we can use numerical methods or approximation techniques to find the solution.

One common numerical method is Newton's method, which is an iterative procedure for approximately solving equations. With an initial guess for x, we can iteratively refine our estimate until we get a sufficiently accurate result.

Another approach is to use a graphing calculator or numerical software to solve the equation. You can plug the equation into the calculator or software and find the approximate values of x that satisfy Equation 1.

Once you find the value(s) of x that satisfies Equation 1, you can substitute it back into the concentration function S = c/x^2 + 9c/(d-x)^2 to find the corresponding concentration value(s). The x-value that corresponds to the minimum concentration value will give you the point on the line joining the smokestacks where the concentration of the deposit is a minimum.

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)