An engineer is designing a pipeline which is supposed to connect two points P an

Question

An engineer is designing a pipeline which is supposed to connect two points P and S. The engineer decides to do it in three sections. The first section runs from point P to point Q, and costs S56 a mile to lay, the second section runs from point Q to point R and costs S52 mile the third runs from point R to point S and costs S48 a mile. Looking at the diagram below, you see that if you know the lengths marked x and y, then you know the positions of Q and R. Find the values of x and y which minimize the cost of the pipeline. 1 Mil 0 miles Preview miles Preview miles

Explanation / Answer

C = 56 sqrt(2² + x²) + 52 sqrt(1² + y²) + 48(10 - x - y)

C/x = 56x/sqrt(4 + x²) - 48 = 0
56² x² = 48²(4 + x²)
(56² - 48²)x² = 48²(4)
x = 48(2)/sqrt(56² - 48²)
x 3.33 miles

C/y = 52y/sqrt(1 + y²) - 48 = 0
52² y² = 48²(1 + y²)
(52² - 48²)y² = 48²
y = 48/sqrt(52² - 48²)
y 2.4 miles

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