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 4 km east of the refinery. The cost of laying pipe is $200,000/km over land to a point P on the north bank and $400,000/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.)
To find the location of point P on the north bank that minimizes the cost of the pipeline, we need to find the distance from the refinery to point P.
Let's assume that the distance from the refinery to point P is x km.
The cost of laying the pipeline over land from the refinery to point P will be $200,000/km multiplied by x km, which gives us a cost of 200,000x dollars.
The cost of laying the pipeline under the river from point P to the tanks will be $400,000/km multiplied by the distance from point P to the tanks, which is 4 km. This gives us a cost of 400,000 * 4 = 1,600,000 dollars.
So, the total cost of the pipeline is given by the sum of the costs over land and under the river:
Total cost = 200,000x + 1,600,000
To minimize the cost of the pipeline, we need to find the value of x that makes the total cost as small as possible.
Now, we can take the derivative of the total cost function with respect to x and set it equal to zero:
d(Total cost)/dx = 200,000
Setting this equal to zero, we get:
200,000x + 1,600,000 = 0
Solving for x, we find:
200,000x = -1,600,000
x = -1,600,000 / 200,000
x = -8
Since it doesn't make sense to have a negative distance, we can conclude that the distance from the refinery to point P should be 8 km.
Therefore, to minimize the cost of the pipeline, point P should be located 8 km from the refinery on the north bank of the river.
To minimize the cost of the pipeline, we need to determine the optimal location point P on the north bank of the river. Let's assume that point P is located x km east of the refinery.
The cost of laying pipe over land is $200,000/km, and the distance from the refinery to point P is x km. So, the cost of laying pipe from the refinery to point P is 200,000x dollars.
The cost of laying pipe under the river is $400,000/km, and the width of the river is 2 km. So, the cost of laying pipe under the river is 400,000 * 2 = 800,000 dollars.
The distance from point P to the storage tanks on the south bank of the river is 4 km. Therefore, the total cost of the pipeline is:
Total cost = Cost over land + Cost under river
= 200,000x + 800,000
To minimize the cost, we need to find the value of x that gives us the minimum total cost. We can do this by taking the derivative of the total cost function with respect to x and setting it equal to zero.
d(Total cost)/dx = 200,000
Setting the derivative equal to zero and solving for x, we get:
200,000 = 0
Since there is no solution to this equation, it means that the total cost function does not have a minimum or maximum. Therefore, we can conclude that the cost of the pipeline will be the same regardless of the location of point P on the north bank of the river.
Therefore, point P can be located anywhere on the north bank of the river, and it will not affect the cost of the pipeline.
answer is 4000 - x meters
c = 2000/meter * (4000 - x) meters + 4000/meter [ sqrt (2000^2 + x^2)]meters
dc/dx = -2000 + 4000 [ .5] 2x / (x^2+4*10^6)^.5
for max or min dc/dx = 0
2000 (x^2+4*10^6)^.5 = 8000 x
(x^2+4*10^6)^.5 = 4 x
x^2 + 4*10^6 = 16 x^2
4 * 10^6 = 165 x^2
x^2 = 400 *10^4 /165 = 2.424242 ...... *10^4
x = 1.56 * 10^2 = 156
4000 - 156 = 3844 meters = 3.844 km
check my arithmetic !