Suppose you are climbing a hill whose shape is given by the equation z = 1500 -

Question

Suppose you are climbing a hill whose shape is given by the equation z = 1500 - 0.005x^2 - 0.01y^2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 1364). The positive x-axis points east and the positive y-axis points north. If you walk due south, will you start to ascend or descend? If you walk northwest, will you start to ascend or descend? At what rate? (Round your answer to two decimal places.) 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? (Round your answer to two decimal places.)

Explanation / Answer

a) Directional derivative in the direction of -j. This is also the negative partial derivative of y.

zy = -.02y, at y = 80, zy = -1.6 and -zy = 1.6. Since it's positive, hill slopes upwards or ascends

Rate = magnitude = 1.6

b) u = 1/2<-1,1> in North-West direction

    Duz(120,80) = <-.01(120),-.02(80)>.u = <-1.2, -1.6>.(-1/2, 1/2) = -2/52 which is negative => descends

    Rate = Magnitude = ||-2/52|| = .28

c) From above, Dz = <-1.2, -1.6> meaning the slope is steepest in SW direction

Rate = Magnitude = Sqrt(1.22 + 1.62) = 2

d) Slope = 2 from above, so angle is tan-1 1/2 = 30 degress.

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