An oil refinery is located on the north bank of a straight river that is 3km wide. a pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 8km east of the refinery. the cost of laying pipe is $500,000/km over land to point p on the north bank and $1,000,000/km under river to the tanks. what is the minimal cost to construct a pipeline from the refinery to the storage tanks?

$6,598,076.21

Explanation:

We can see the river (sort of) that is 3 km wide and the storage tanks on the other side of the river 8 km apart.

Laying pipes under (across) the river costs 1,000,000 the km & laying pipes over land costs 500,000 per km.

The distance across the river can be found by using Pythagoras Theorem. A side is 3 km the other is unknown, so we call it X. And it is equal to:

And we multiply it by 1,000,000; the cost of laying pipe under the river, the we get:

The distance over the land is (8-x), as we can see in the drawing. So we multiply it by its cost, 500,000. And we get 500,000(8-x).

So the cost function f(x) would be:

And we plug square root of 3 in the original cost function  ad we get

u = 30

continuing your method

= +

subtract from both sides

= - thus

= -

= ⇒ u = 30