Question

Your house lies on the surface z = f(x,y) = 2(x/50)^2 – (y/50)^2 directly above the point (200,150) in the xy-plane. x, y, and z are measured in meters. a) How high above the xy-plane do you live? b) What is the slope of your lawn when you look from your house directly toward the z-axis (that is along the vector -200i – 150j)? Express the slope as an angle in degrees! c) When you wash your care in the driveway, on this surface above the point (200,150), which way does the water run off? Give the answer as a two directional vector. (Water runs off in direction of steepest decent)

Answer 1

for part a) plug x,y in and find z

part b) you'll need to create a level surface \(\psi = 2({ x\over 50})^2 – ({y \over 50})^2 - z \) and compute \(\nabla \psi\) at point (200, 150, whatever z is). you then want to dot that with the vector \(-200 \hat i – 150 \hat j\) or better the unit vector in that direction

for c) i think you just need the gradient of f(x,y) a that pount to find the direction of greatest change in z.

hope that helps :-)