1. ## Max/min Help!

This is yr 12 maths b.A bushwalker can walk at 5km/h through clear land and 3km/h through bushland. If she has to het from point a to point b following a route indicated in the image, find the value of x so that the route is covered in a minimum time. (Note: time = distance/speed)

Image: http://img294.imageshack.us/my.php?i...ntitledtp0.jpg

2. Hello, mmulv2!

The set-up is the hard part . . . I'll leave the rest to you.

A bushwalker can walk at 5 km/h through clear land and 3 km/h through bushland.
If she has to get from point A to point B following a route indicated in the image,
find the value of x so that the route is covered in a minimum time.
Code:
                        * B
|
| 3-x
|
* C
*  |
____     *     |
√x²+4  *        | x
*           |
*              |
A * - - - - - - - - * D
2

She will walk from A to C through bushland.
. . From right triangle $\displaystyle CDA$, we have: .$\displaystyle AC \:=\:\sqrt{x^2+4}$ km.
At 3 km/h, it will take her: .$\displaystyle T_1\;=\;\frac{\sqrt{x^2+4}}{3}$ hours.

Then she will walk from C to B through clear land.
. . At 5 km/h, this will take her: .$\displaystyle T_2 \;=\;\frac{3-x}{5}$ hours.

Hence, her total time is: .$\displaystyle T \;=\;\frac{1}{3}(x^2+4)^{\frac{1}{2}} + \frac{3}{5} - \frac{1}{5}x$

. . and that is the function you must minimize.