suppose you are climbing a hill whose shape is given by the equation z=1000-0.00

Question

suppose you are climbing a hill whose shape is given by the equation z=1000-0.005x^2 -0.01y^2, where x,y, and z are measured in meters, and you re 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 diection begin?

Explanation / Answer

The gradient is (-.01x,-.02y)
At (60,40,996), this is (-.01(60),-.02(40)) = (-.6,-.8)

a) Due south is (0,-1) (north is the positive y direction, so this is the due south direction)
(0,-1) . (-.6,-.8) = .8

You will be ascending. The slope is .8

b) The direction northwest is (-1/2,1/2)

(-1/2,1/2) . (-.6,-.8) = -.2/2 = -1/10 2

This is a descent direction. The slope is -1/10 2

c) The slope is largest in the (-.6,-.8) direction. Conveniently, this is already a unit direction.

Thus, the increase is (-.6)^2+(-.8)^2 = 1

Thus, there is one unit of ascent for each unit in the (-.6,-.8) direction. This is 45 degrees.

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