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 6 km east of the refinery. The cost of laying pipe is $400,000 per km over land to a point P on the north bank and $800,000 per 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.)

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 denote the distance from the refinery to point P as x kilometers.

The cost of laying pipe over land from the refinery to point P is $400,000 per km, so the cost of this segment is 400,000x.

The cost of laying pipe under the river from point P to the storage tanks is $800,000 per km, and the width of the river is 2 km. Therefore, the cost of this segment is 800,000 * 2 = 1,600,000.

The total cost of the pipeline can be expressed as the sum of the cost over land and the cost under the river:

Total Cost = Cost over land + Cost under the river
Total Cost = 400,000x + 1,600,000

To minimize the cost of the pipeline, we need to find the value of x that minimizes the total cost. We can achieve this by taking the derivative of the total cost with respect to x and setting it equal to zero.

d(Total Cost)/dx = 400,000 = 0

Solving for x, we find:

400,000x = 1,600,000
x = 4

Therefore, to minimize the cost of the pipeline, point P should be located 4 km from the refinery.

To minimize the cost of the pipeline, we need to determine the optimal location on the north bank where the pipeline should start (point P).

Let's assume that point P is located x kilometers east of the refinery.

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 6 km - x km (since the storage tanks are 6 km east of the refinery).

The cost of laying pipe over land is $400,000 per kilometer, so the cost of laying the pipe from the refinery to point P is 400,000 * x dollars.

The cost of laying pipe under the river is $800,000 per kilometer, so the cost of laying the pipe from point P to the storage tanks is 800,000 * (6 - x) dollars.

Therefore, the total cost C of laying the entire pipeline can be expressed as:

C = 400,000x + 800,000(6 - x)

Now, to find the optimal location for point P, we need to minimize the total cost C.

To do this, we can find the derivative of C with respect to x and set it equal to zero.

dC/dx = 400,000 - 800,000

Setting this equal to zero:

400,000 - 800,000 = 0

Simplifying, we find:

-400,000 = 0

Since -400,000 does not equal zero, there is no critical point.

However, since we are dealing with a linear cost function, we know that the minimum or maximum occurs at one of the endpoints.

In this case, the endpoints are x = 0 (the refinery) and x = 2 (the midpoint of the river).

To determine which endpoint gives the minimum cost, we can evaluate the cost function at both endpoints.

C(0) = 400,000 * 0 + 800,000 * (6 - 0) = 4,800,000

C(2) = 400,000 * 2 + 800,000 * (6 - 2) = 4,000,000

Comparing C(0) and C(2), we see that C(2) gives the minimum cost.

Therefore, we conclude that the optimal location for point P is 2 kilometers east of the refinery.

So, the pipeline should start at a point P located 2 kilometers east of the refinery in order to minimize the cost.

Let the pipe go from R (refinary) to P over a distance of 6-x km.

The distance (under water) from P to S (storage) is therefore √(x²+2²).
The total cost, C
C = C1(6-x) + C2 (√(x²+2²))
where C1=cost on land
C2 = cost under water, /km

Differentiate C(x) with respect to x and equate to zero for a local minimum.

Solve for x.

x=4.4