Answer:
We can use the Pythagorean theorem to find the distance between the oil well and the refinery, then use that distance to find the minimum cost path for the pipe.
First, let's label the oil well as point O, the point on the coast directly south of the well as point E, and the refinery as point R. We can then use the Pythagorean theorem to find the distance between the oil well and the refinery:
OE = 3 km
ER = 4 km
OR^2 = OE^2 + ER^2
OR^2 = 9 + 16
OR = 5 km
Now that we know the distance between the oil well and the refinery, we can use that distance to find the minimum cost path for the pipe. Since the cost of laying pipe in the water is greater than the cost of laying pipe on the coast, it will be cheaper to lay as much pipe on the coast as possible.
To lay as much pipe on the coast as possible, the pipe should follow the path ER, then turn and follow the path OR to the oil well. This will minimize the cost of the pipe because it will minimize the amount of pipe that must be laid in the water.
The total cost of the pipe will be the cost of laying the pipe on the coast plus the cost of laying the pipe in the water. Since the cost of laying pipe on the coast is $300,000 per kilometer and the cost of laying pipe in the water is $500,000 per kilometer, the total cost of the pipe will be:
Cost = (ER * $300,000/km) + (OR * $500,000/km)
Cost = (4 km * $300,000/km) + (5 km * $500,000/km)
Cost = $1,200,000 + $2,500,000
Cost = $3,700,000
Answer:
Step-by-step explanation:
You want to know the least-cost path of a pipeline from 3 km offshore to 4 km east of the closest point on shore. The cost of pipe in the water is $0.5M/km, and onshore is $0.3M/km.
The solution in the general case for this kind of question is that the sine of the angle between the route directly to shore and the least-cost route is the ratio of the onshore cost to the in-water cost.
sin(α) = (0.3M/km)/(0.5M/km) = 3/5
The distance from the closest point onshore (E) to the least-cost onshore point is the product of the distance from shore and the tangent of this angle:
distance downshore = (3 km)·tan(arccos(3/5)) = 2.25 km
Since the refinery is 4 km downshore, this point is 1.75 km from the refinery.
The pipe should follow a direct path underwater from the well to a point 1.75 km west of the refinery, then downshore the remaining 1.75 km to the refinery.
If x represents the distance from the refinery to the point where the pipeline comes ashore, the cost of the route in millions will be ...
c = 0.5√(3² +(4-x)²) +0.3x
where the square root function is calculating the underwater length of the pipeline using the Pythagorean theorem.
This cost function will be minimized when its derivative with respect to x is zero.
c' = 0.5(1/2)(2x -8)/√(x² -8x +25) +0.3 = 0
0 = 0.5(x -4) +0.3√(x² -8x +25) . . . . . . numerator of the combined terms
4 -x = 0.6√(x² -8x +25) . . . . multiply by 2, add 4-x
16 -8x +x² = 0.36x² -2.88x +9 . . . . . . square both sides
0.64x² -5.12x +7 = 0 . . . . . subtract the right side
x = (5.12 -√(5.12² -4(0.64)(7)))/(2(0.64)) = 2.24/1.28 = 1.75
The route with lowest cost comes ashore 1.75 km from the refinery.
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Additional comment
The cost of the pipeline will be ...
c = 0.5√(9 +(4-1.75)²) +0.3(1.75) = 0.5√14.0625 +0.525
c = 2.4 . . . . million dollars
The attached graph shows the cost function has a minimum of 2.4M at a distance of 1.75 km from the refinery. The cost is only 2.5M if the pipe runs directly from the well to the refinery (x=0).
