The surface of mountain is modeled by the equation h(x,y)=5000 -0.001x^2-0.004y^2. A mountain climber is at the point (500, 300, 4390). In what direction should the climber move in order to ascend at the greatest rates ? Give direction as a vector only i & j components needed.
From that, give the direction that climber should move so that his altitude does not change (h doesn't change). The climber is still assumed to be at the point (500,300,4390).
grad(h) = (D_x h(x,y), D_y h(x,y)) = (-0.002 x, -0.008 y)
So the direction of steepest ascent is parallel to:
(-0.002*500, -0.008*300) = (1, 2.4)
reducing this to a unit vector gives:
u = (1, 2.4)/sqrt(1+2.4^2) ~= (0.3846, 0.9231) = 0.3846 i + 0.9231 j
The rate of ascent in direction v is v.grad(h), so to maintain a constant heigh the
climber must move off in a direction such that v.grad(h) = 0, so if we put:
v = -0.9231 i + 0.3846 j
this will do the job.
RonL