Suppose you are climbing a hill whose shape is given by the equation $z=1300-0\mathrm{.}005x^2-0\mathrm{.}01y^2,$ where $x,y,$ and $z$ are measured in meters, and you are standing at a point with coordinates $(60,40,1266).$ 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?

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At what rate?

vertical meters per horizontal meter

$($b$)$ If you walk northwest, will you start to ascend or descend?

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At what rate? $($Round your answer to two decimal places.$)$

vertical meters per horizontal meter

$($c$)$ In which direction is the slope largest?

What is the rate of ascent in that direction?

vertical meters per horizontal meter