Hello, warsatan!
An offshore oil well is 2 kilometers off the coast.
The refinery is 4 kilometers down the cost.
If laying pipe in the ocean is twice as expensive as on land,
what path should the pipe follow in order to mimize the cost?
Code:
. . . . W
. . . . * The oil well is at W.
. . . . | \ The refinery is at R.
. . . . | \
. . . 2 | \ The pipe is laid underwater to B
. . . . | \ then along the coast to R.
. . . . | \
. . . - + - - - - - * - - - - - *
. . . . A . . x . . B . .4-x. . R
Let \(\displaystyle p\) = price for laying pipe on land (per kilometer).
Then \(\displaystyle 2p\) = price of laying pipe underwater.
From right triangle WAB, we get:
.\(\displaystyle WB\:=\:\sqrt{x^2\,+2^2}\) km of underwater pipe.
. . This will cost:
.\(\displaystyle 2p\sqrt{x^2 + 4}\) dollars.
Let \(\displaystyle x\,=\,AB\)
There will be \(\displaystyle 4 - x\) km of pipe laid along the shore.
. . This will cost:
.\(\displaystyle p(4 - x)\) dollars.
The total cost is:
.\(\displaystyle C\:=\:2p(x^2\,+\,4)^{\frac{1}{2}}\,+\,p(4\,-\,x)\)
And
that is the function you must minimize . . .
