Question

Suppose you are climbing a hill whose shape is given by the equation z = 1400 − 0.005x2 − 0.01y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (100, 120, 1206). 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: You start to ascend

Explanation:

Since you are walking due south and the positive y-axis points north, the y-coordinate decreases.

Let 

$y'$ = y-coordinate after walking due south
$z'$ = z-coordinate (or the height) after walking due south

So, the point where you stand after walking due south has a coordinate of $(100, y', z')$. Moreover,

$z' = 1,400 - 0.05(100)^2 - 0.01(y')^2 \\ \boxed{z' = 1350 - 0.01(y')^2}$

Since the y-coordinate decreases after walking due south and $z' = 1350 - 0.01(y')^2$,

$y' \leq 120\\ \Rightarrow (y')^2 \leq 120^2 = 14,400\\ \Rightarrow -0.01(y')^2 \geq -0.01(14,400) = -144\\ \Rightarrow z' = 1,350 - 0.01(y')^2 \geq 1,350 - 0.01(14,400) = 1,206\\ \Rightarrow \boxed{z' \geq 1,206}$

So, the height after walking due south exceeds 1,206 meters, which is the previous height before walking due south. Therefore, if you walk due south from point with coordinates (100, 120, 1206), you are ascending.