Suppose that the Oil Refinery is located 1k north of the river that is 2km wide. A pipeline is to be contructed from the refinery to storage tanks located on the south bank of the river 6km east of the refinery. The cost of laying pipe is $400,000km over land to a point P on the north bank and $800,000km under the river to the tanks. Where should P be located to minimize the cost of the pipeline?
By Pythagorean Theorem, the distance between the refinery to point P is ...
d1=√(6−x)2+12=√x2−12x+37
And, the distance between point P to the storage is...
d2=√x2+4
Therefore, the total cost would be...
cost = 400,000√x2−12x+37+800,000√x2+4
If we take the derivative of cost, we have...
c′(x)=400,000(2x−122√x2−12x+37)+800,000(2x2√x2+4)
when c′(x)=0
0=400,000[\cancel2(x−6)\cancel2√x2−12x+37+2(\cancel2x\cancel2√x2+4)]0=x−6√x2−12x+37+2x√x2+4
Solving for x, we have...
x=1.12 km
If we evaluate the cost at x=0, x=6km and x=1.12km, then...
c(0)=400,000(√02−12(0)+37)+800,000(√02+4)c(0)=$4,033,105.012c(6)=400,00(√62−12(6)+37)+800,000(√62+4)c(0)=$5,459,644.256c(1.12)=400,000(√(1.12)2−12(1.12)+37)+800,000(√(1.12)2+4)c(1.12)=$3,826,360.414
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