7. (15 points) A hiker is walking on a mountain path. The surface of the mountai
ID: 2891109 • Letter: 7
Question
7. (15 points) A hiker is walking on a mountain path. The surface of the mountain is modeled by z = 100 + 6r-4x2-3r with measurements in t. The positive z axis points to the East direction (1,0), the negative s-axis points West (-1,0), the positive y-axis points North (0,1) and the negative y-axis points South (0, -1). (a) (4 points) Suppose the hiker is now at the point P(1,2,90) heading West. Is she ascending or descending? Explain your answer. (b) (3 points) Suppose the hiker who is now at P(1, 1,99) but is now heading North with a map speed of 3 feet per second. (Note: the hiker's "map speed" is not the speed of the hiker along the mountain's surface, which will be greater.) At what speed in feet per second is the elevation of the hiker changing? Remember that speed is a positive number. (c) (4 points) When the hiker is at the point Q(2,0,96), in which direction on her map should she initially head to ascend most rapidly? Why? (d) (4 points) When the hiker is at the point Q(2,0,96), in which two directions on her map can she initially head to neither ascend nor descend (to keep traveling at the same beight)? Explain your answer.Explanation / Answer
7)
given z=f(x,y)=100+6x-4x2-3y2 .
gradient,f= 6-8x,-6y
(a)
at P(1,2,90)
f= 6-8*1,-6*2
f= -2,-12
u=-1,0
Duf =f.u
Duf =-2,-12.-1,0
Duf =(-2*-1)+(-12*0)
Duf =2
Duf >0
directional derivative greater than zero.so she is ascending
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(b)
at P(1,1,99)
f= 6-8,-6
f= -2,-6
Duf =-2,-6.0,1
Duf =(-2*0)+(-6*1)
Duf =-6
rate of change of elevation=|-6|*3
rate of change of elevation=18 feet per seond
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(c)
at Q (2,0,96)
f= 6-8*2,-6*0
f= -10,0 is the direction to ascend rapidly
hiker should go west
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d)
let the direction in which there is no ascend/descend be a,b
f.a,b=0
-10,0.a,b =0
=>-10a +0=
=>a=0
direction in which there is no ascend/descend be 0,b
she should head eith north or south
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