- #1
Painguy
- 120
- 0
Homework Statement
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 the following equation, where k is a positive constants that depends on the quantity of smoke each stack is emitting.
[itex]S=((64k)/(x^2)) + (k/((20-x)^2))[/itex]
Homework Equations
The Attempt at a Solution
1st I take the derivative
[itex]((-128k(20-x)^3))+2kx^3/(x^3)(20-x)^3))[/itex]
Then i look for critical points
x^3=0
x=0
20-x=0
x=20
The third one is where i get stuck
(-128k(20-x)^3))+2kx^3=0
k(-128(-x^3 + 60x^2 -1200x +8000) + 2x^3)=0
(-128(-x^3 + 60x^2 -1200x +8000) + 2x^3)=0
128x^3-7680x^2 +153600x - 1024000 + 2x^3=0
130x^3-7680x^2 +153600x - 1024000=0
130x^3-7680x^2 +153600x = 1024000
how would I solve for this? It's a bit embarrassing on my part, but I never ran into a situation before where I had to find the root of a cubic function.