Suppose you are climbing a hill whose shape is given by the equation below, wher

Question

Suppose you are climbing a hill whose shape is given by the equation below, where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 964). The positivex-axis points east and the positive y-axis points north.

If you walk northwest, will you start to ascend or descend

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

What is the rate of ascent in that direction?

At what angle above the horizontal does the path in that direction begin? (Round the answer to two decimal places.)
°

Explanation / Answer

Compute the gradient z = <-0.01x, -0.02y>.

(a) Walking south, the unit vector in this direction is u = <0, -1>.
So, D_u (120, 80)
= z(120, 80) · u
= <-1.2, -1.6> · <0, -1>
= 1.6

Since this is positive, you are ascending.

(b)

Walking northwest (angle 3/4 with the horizontal), the unit vector in this
direction is u = <cos 3/4, sin 3/4> = <-2/2, 2/2>.
So, D_u (120, 80)
= z(120, 80) · u
= <-1.2, -1.6> · <-2/2, 2/2> = -0.28

Since this is negative , you are descending.

The slope of largest direction is in the direction of the gradient.

==> u = z(120, 80) / ||z(120, 80)|| = <-1.2, -1.6> / 2

What is the rate of ascent in that direction?

z(120, 80) · u = <-1.2, -1.6> · <-1.2, -1.6>/ 2 = 2

At what angle above the horizontal does the path in that direction begin?

Using the unit vector u
= arctan(2)
=> =63.43 degrees

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