Hello again, lmao!
Here's #5 . . .
5. A small resort is situated on an island.
The closest point on the mainland is a point P which is exactly 3 miles from the resort.
The closest source of fresh water is 10 miles down the shoreline from P.
The resort is planning to lay pipe to bring water from that source.
It will run along the shore for some distance and then turn and cross the water to the resort.
However, running pipe underwater costs 2.4 times as much as running pipe down the shore.
At what distance down the shore from P should the pipe turn and head for the resort?
| * ____
3 | *-√x²+9
* - - - - - * - - - - - - - *
P x Q 10-x W
The resort is at R, 3 miles from P on the shore.
The water source is at W: PW = 10.
They will run the pipe from P to point Q on the shore,
. . where: PQ = x, and: QW = 10 - x
Suppose it costs k dollars/mile to lay pipe on land.
. . The cost of the land pipe is: .k(10 - x) dollars.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _____
From right triangle RPQ, we have: RQ .= .√x² + 9 miles.
At 2.4k dollars/mile to lay underwater pipe,
. . the cost is: .2.4k√x² + 9 dollars.
Hence, the total cost is: .C .= .2.4k(x² + 9)^½ + 10k - kx
Can you finish it now?