# Problem: Suppose you are climbing a hill whose shape is given by the equation z=1000-0.005x2-0.01y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (60,40,966). 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? At what rate?(b) If you walk northwest, will you start to ascend or descend? At what rate?(c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

###### FREE Expert Solution

We have teh equation, z = 1000 - 0.005x2 - 0.01y2

(a)

We need to strat by finding directional derivatives:

$\begin{array}{rcl}\frac{\mathbf{\partial }\mathbf{z}}{\mathbf{\partial }\mathbf{x}}& \mathbf{=}& \frac{\mathbf{\partial }\mathbf{\left(}\mathbf{1000}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{005}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{01}{\mathbf{y}}^{\mathbf{2}}\mathbf{\right)}}{\mathbf{\partial }\mathbf{x}}\\ & \mathbf{=}& \mathbf{-}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{\left(}\mathbf{0}\mathbf{.}\mathbf{005}\mathbf{x}\mathbf{\right)}\end{array}$

∂z/∂x = - 0.01x

Similarly,

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###### Problem Details

Suppose you are climbing a hill whose shape is given by the equation z=1000-0.005x2-0.01y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (60,40,966). 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? At what rate?
(b) If you walk northwest, will you start to ascend or descend? At what rate?
(c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?