The town of Salter Path is investigating purchasing water from the town of Moreh
ID: 3210893 • Letter: T
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/StfF2 Can someone show step by step how to get the answer please?Explanation / Answer
There's no drawing at that site.
Glanced at Salter Path, Bogue Sound, Morehead City on Google Maps.
I'm guessing that the distance from Morehead City to point C is not directly across the sound. If that's the case, then its distance will be the hypotenuse of a right triangle.
The distance from C to Salter Path will be a straight line.
Your costs are applied to the distances: $500000/mile for the piece from point A to point C and $300k/mile for the piece from point C to Salter Path.
Your distance from point C to Salter Path will be a fraction of the length of the strip of land directly across the sound from point A. Where the fractions are x and (L - x),
with x being the piece from point C to Salter Path, your underground piping cost is $300k(x).
For the underwater part, (L-x) is one leg of the triangle whose hypotenuse is the distance between points A and C. I expect you were given the distance across the sound, so that's the other leg.
The underwater cost is will be something like $500000((L-x)2 + r2)), where r is the distance across the sound.
Differentiate the cost (underwater + underground) with respect to x. Set dC/dx = 0 and solve for x. That will give you the location for point C for minimum cost. Test by plugging in locations for C that are closer to and further from Salter Path than x.
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