Question

Suppose you are climbing a hill whose shape is given by the equation z = 1300 − 0.005x2 − 0.01y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 1164). The positive x-axis points east and the positive y-axis points north. (a) If you walk due south, will you start to ascend or descend?

Answer 1

Answer:

  ascend

Step-by-step explanation:

There are a couple of ways to figure this. One is to look at the partial derivative ...

  $\dfrac{\partial z}{\partial y}=-0.02y$

The sign of it is negative, so when y decreases, z increases. Going south means decreasing the y-coordinate, so moving in that direction will cause you to ascend.

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Another way to figure this is to evaluate z for some small movement in the southerly direction. In the attached, we find that moving 1 m south to (120, 79, 1165.59) causes an elevation increase of about 1.6 m. Walking south causes you to ascend.

Note that this is consistent with our first result, as -0.02y = -0.02(80) = -1.6. So a change in y of -1 should cause a change in z of about (-1.6)(-1) = 1.6.