math question about directional derivatives?

Question

suppose you are on a hill described by z = 1000 - 0.01x^2 - 0.02y^2 where x, y, z are measured in meters.
you are standing at point (50, 80, 847), and the positive x axis points east and the positive y axis points north.

if you walk due south, will you ascend or descend? at what rate?

I'm really not sure, but I got an answer of ascend at rate of 3.2 meters.
please help, thanks

Answer 1

If you're at the point (50, 80, 847) and you walk south, you're in the plane x = 50.
Set x equal to 50:
z = 1000 - 0.01x^2 - 0.02y^2
z = 1000 - 0.01(50)^2 - 0.02y^2
z = 975 - 0.02y^2
This is what the crossection of the hill looks like in the plane x = 50:

http://s236.photobucket.com/albums/ff177/jsardi56/?action=view¤t=planexequalsfifty.jpg

You can see that as you go north you're going downhill.
Take the derivative with respect to y and set y = 80:
z = 975 - 0.02y^2
dz/dy = - 0.04y = - 0.04(80) = -3.2 descend