In oil pipeline construction, the cost of pipe to go underwater is 60% more than the cost of pipe used in dry-land. A pipeline comes to a river that is 1KM wide at point A and must be extended to a refinery, R, on the other side, 8KM down the river. Find the best way to cross the river (Assuming it is straight) so that the total cost of the pipe is kept a minimum. (give you answer correct to one decimal place)

Question

anonymous · Answer

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anonymous · Answer

this is the same question and answer but i dont get it can someone explain it to me http://ca.answers.yahoo.com/question/index?qid=20100326222123AAOmh5c

zehanz · Answer

Well maybe I can help you with my view of the situation:

zehanz · Answer

Now we need a function to calculate the cost. I call this function C.
The variable of this function is x. It is (see drawing) the part of the river bank where there is no pipeline, so the other part is 8 - x.
Then the part crossing the river has length $\sqrt {x^2+1}$.

zehanz · Answer

Now for the actual cost: we don't know that, only that it is 60% more expensive on water than on land.
Suppose the cost of laying the pipeline on land is a dollars per km, then it is 1.6a on water, giving the function:

$C(x)=a(8-x)+1.6a\sqrt{x^2+1}$.

Because of the situation you described, x is restricted to values between 0 and 8.

zehanz · Answer

We have to solve C'(x)=0:

$C'(x)=-a+1.6a\cdot \dfrac{2x}{2\sqrt{x^2+1}}=-a+\dfrac{1.6ax}{\sqrt{x^2+1}}=0$
Dividing by a gets rid of it (so it is unimportant to know the actual value of a after all) and rearrange:

$\dfrac{1.6x}{\sqrt{x^2+1}}=1$

zehanz · Answer

This leads to:
 
$1.6x=\sqrt{x^2+1}$.

Squaring both sides:

$2.56x^2=x^2+1$, so $1.56x^2=1$ and $x^2=\dfrac{1}{1.56}$.

Taking the positive root (remember x is between 0 and 8) gives $x\approx 0.8$.

If you are very serious about this, a sign scheme of C' would prove that for this value of x the function C has indeed a minimum.
Here is a graph of C (with a = 0.5, could be any positive number). It shows a minimum for x = 0.8.

zehanz · Answer

Now to give a clear answer to the original question: Instead of crossing the river at a straight angle, make the pipeline so that it arrives on land about 800m (0.8km) in the direction of R. See drawing.

zehanz · Answer

I hope you understand this explanation!