Calc 3, web assign, Thanks. k Your Suppose you are climbing a hill whose shape i

Question

Calc 3, web assign, Thanks.

k Your Suppose you are climbing a hill whose shape is given by the equation z points east and the positive y-axis points north. 2000 - 0.005x2-0.012, where x, y, and z are measured in meters, and you are standing at a point with coordinates (100, 120, 1806). The positive x-axls (a) If you walk due south, will you start to ascend or descend? O ascend O descend At what rate? vertical meters per horizontal meter (b) If you walk northwest, witl you start to ascend or descend? O ascend O descend 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 vertical meters per horizontal meter At what angle above the horizontal does the path in that direction begin? (Round your answer to two decimal places.) Need Helo? IRead it Talk to a Tutor

Explanation / Answer

given z=f(x,y)= 2000-0.005x2-0.01y2

gradient,f= 0-0.005*2x2-1-0,0-0-0.01*2y2-1

f= -0.01x,-0.02y

at (x,y,z)=(100,120,1806)

f= -0.01*100,-0.02*120

f= -1, -2.4

(a)

unit direction down south is u=0,-1

Duf = -1, -2.4.0,-1

Duf = (-1*0) +(-2.4*-1)

Duf = 2.4

ascend

2.4 vertical meters per horizontal meter

(b)

unit direction down northwest is u=-1/2,1/2

Duf = -1, -2.4.-1/2,1/2

Duf = (1 -2.4)(1/2)

Duf = (-1.4/2)

Duf =-0.99

descend

0.99 vertical meters per horizontal meter

(c)

slope is maximum in direction of vector f= -1, -2.4

|f|=[(-1)2+(-2.4)2]=6.76 =2.6

unit vector in direction of largest slope = f/|f|

unit vector in direction of largest slope = -1/2.6, -2.4/2.6

unit vector in direction of largest slope = -5/13, -12/13

rate of ascent = |f| =2.6

2.6 vertical meters per horizontal meter

angle above the horizontal =tan-1(|f|)

angle above the horizontal =tan-1(2.6)

angle above the horizontal =68.96o

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