The town of Salter Path is investigating purchasing water from the town of Moreh

Question

The town of Salter Path is investigating purchasing water from the town of Morehead City. The main expense invilved will be the construction of a pipeline across Bogue Sound. The cost of constructing pipeline underwater is estimated at $500,000 per mile, while the cost of constructing pipeline underground is estimated at $300,000 per mile. Find the exact value of the distance from point A to the variable point C so that if pipeline is run underwater from Morehead City to C and then underground from C to Salter Path, then the total cost of constructing the pipeling will be minimized. Verify your answer with an appropriate test. Picture http://imgur.com/zM89z Can someone show step by step how to get the answer please?

Explanation / Answer

from the figure the distance from morehead city to point C is =sqrt(x^2+2^2)=sqrt(x^2+4)..........distance from C to salter path is 9-x.....therefore cost funtion is given by c(x)=500000*sqrt(x^2+4)+(9-x)*300000.....for this cost to be minimum d(c(x))/dx=0............we get 2*x/(2*sqrt(x^2+4))*500000-300000=0......solving we get x=1.5....at which the min.cost is c(1.5)=3500000$

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