Finding the Minimum (Word Problem)
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=c/(x^2)+k/((d−x)^2)
where c and k are positive constants which depend on the quantity of smoke each stack is emitting. If k=2c, find the point on the line joining the stacks where the concentration of the deposit is a minimum.
I plugged in 2c for k and then differentiated to get:
Then I figured that I should set it equal to zero and solve for x, which I tried to do but did so incorrectly.
Can someone give me a hand? Am I going about this the right way? Did I make a mistake when I took the derivative? Should I be setting the derivative equal to zero and solving for x? What should x be then? (Headbang)