Hello, ShawnC!

A hunter is at a point on a straight river bank running east-west.

He wants to get to his cabin located 3 miles north and 8 miles west.

He can travel 5 mph on the river but only 2 mph on the rocky land.

How far upriver should he go in order to reach the cabin in minimum time? The set-up is easier than you think . . . Code:

: - - - - 8 - - - - :
C 8-x P x A
o - - - - - o - - - - o
* |
_____ * | 3
√x²+3² * |
* |
o
H

The hunter is at $\displaystyle H.$

He wants to get to his cabin $\displaystyle C.$

He crosses the river to point $\displaystyle P$

then hikes the remaining distance to $\displaystyle C.$

The distance across the river is: ,$\displaystyle HP \:=\:\sqrt{x^2+9}$ miles.

. . At 5 mph, this will take him: .$\displaystyle \frac{\sqrt{x^2+9}}{5}$ hours.

The distance he hikes on rocky land is: .$\displaystyle PC \;=\;8-x$ miles.

. . At 2 mph, this will take him: .$\displaystyle \frac{8-x}{2} $ hours.

Hence, his total time is: .$\displaystyle T \;=\;\tfrac{1}{5}\left(x^2+9\right)^{\frac{1}{2}} + 4 - \tfrac{1}{2}x$ hours.

And *that* is the function we must minimize.