Minimizing pipeline cost
A utility company must run a pipeline from a station on one side of a straight river to a plant on the other side. The river is miles wide, and the plant is located miles downstream from the point directly across the river from the station. The pipeline will run underwater in a straight line from the station to some point on the opposite bank, and then along the bank (over land) to the plant. Let be the distance (in miles), measured along the bank, from the point directly across from the station to the point where the pipeline comes ashore. Underwater pipe costs \6$4$ million per mile.
The total cost of the pipeline (in millions of dollars) as a function of is Here is the distance along the bank from the point directly opposite the station to where the pipe reaches shore, and . Find the value of that minimizes the total cost .
Hint
Take the derivative , set it equal to zero, and solve for . Since always, be careful when clearing the square root — check that your solution satisfies the original equation (i.e., has the correct sign) and lies in the domain .
Step-by-step solution
Differentiate to get . Setting gives , so , requiring .
Squaring both sides: , so , giving and . Since for and for (or by checking ), this critical point is a minimum, and it lies in , so it is the answer.
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