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(60, 40, 1666). 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?
ascend
descend


At what rate?

___________

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


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

_____________


(c) In which direction is the slope largest?
(-0.6, -0.8)
(0.6, 0.8)
(0.6, -0.8)
(-0.6, 0.8)


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.)

__________________ °

Answer 1

(a) Walking south, the unit vector in this direction is u = <0, -1>.
So, D_u (300, 200)
= ?z(300, 200) · u
= <-3, -0.4> · <0, -1>
= 0.4.

Since this is positive, you are ascending.

(b) Walking northwest (angle 3p/4 with the horizontal), the unit vector in this
direction is u = <cos 3p/4, sin 3p/4> = <-v2/2, v2/2>.
So, D_u (300, 200)
= ?z(300, 200) · u
= <-3, -0.4> · <-v2/2, v2/2>
= (1.3) v2.

Since this is positive, you are ascending.

(c) The slope of largest direction is in the direction of the gradient.
==> u = ?z(300, 200) / ||?z(300, 200)|| = <-3, -0.4> / v9.16.

(d) From part (c), ?z(300, 200) · u = <-3, -0.4> · <-3, -0.4> / v9.16 = v9.16.

(e) Using the unit vector u from (c) and (d):
? = arctan(y/x) + p = arctan[(-0.4)/(-3)] + p, since x < 0
==> ? = 3.274... radians.