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 3 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.)

P is exactly 3km east from the oil refinery.

Step-by-step explanation:

Let's d be the distance in km from the oil refinery to point P. So the horizontal distance from P to the storage is 3 - d and the vertical distance is 2. Hence the diagonal distance is:

So the cost of laying pipe under water with this distance is

And the cost of laying pipe over land from the refinery to point P is 400000d. Hence the total cost:

We can find the minimum value of this by taking the 1st derivative and set it to 0

We can move the first term over to the right hand side and divide both sides by 400000

From here we can square up both sides

d = 3

So the cost of pipeline is minimum when P is exactly 3km east from the oil refinery.

there is no question content here, but you for the free points! : d

if x = 1, then the answer is 3.5