Question

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 7 km east of the refinery. The cost of laying pipe is $400,000/km over land to a point P on the north bank and $800,000/km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Round your answer to two decimal places.)

Answer 1

To minimize the cost of the pipeline, the optimal location for point P should be determined by solving the equation 400,000 * x + 800,000 * (7 - x) = total cost, where x is the distance from the refinery to point P.

Step-by-step explanation:

To minimize the cost of the pipeline, we need to find the optimal location for point P on the north bank of the river. Let's assume that the distance from the refinery to point P is x km. The distance from point P to the storage tanks on the south bank is then 7 - x km.

The cost of laying pipe over land is $400,000/km, so the cost of laying pipe from the refinery to point P is 400,000 * x dollars. The cost of laying pipe under the river is $800,000/km, so the cost of laying pipe from point P to the storage tanks is 800,000 * (7 - x) dollars.

The total cost of the pipeline is the sum of these two costs. To minimize this cost, we need to find the value of x that minimizes the total cost.

To solve for x, we can set up an equation:

400,000 * x + 800,000 * (7 - x) = total cost

By simplifying and solving this equation, we can find the optimal location for point P.

Answer 2

To minimize the cost, we take the straight distance from the refinery to the other side of the river as 2 km. Also, the 7 km will be the distance that has to be traveled by the pipeline in land. The total cost, C, is therefore,
total cost = (2 km)($800,000/km) + (7 km)($400,000 /km)
total cost = $4,400,000
Thus, the total cost of the pipeline is approximately $4,400,000.00.