Drifting snow in a parking lot has collected to height z = 6 + 3x - y^2 + xy inc

Question

Drifting snow in a parking lot has collected to height z = 6 + 3x - y^2 + xy inches, with distances x and y measured in feet. (a) Find the rate at which the height of snow, in inches/foot, is changing at the point (3, 1) in the direction towards the point (4, 4). (b) In what direction is the height of the snow increasing most rapidly at the point (3, 1)? (c) What is the rate of change in the direction of maximum change at the point (3, 1)? (d) The position on the ground below, at time t seconds, of a bug crawling on the snow is (2t, t^2 - 3). What is the time t at which the bug will be at position (4, 1)? (e) What is the rate dz/dt, in inches/second, at which the bug experiences a change in height when it is at (4, 1)?

Explanation / Answer

z = 6 + 3x - y2 + xy

grad(z) = ( 3+ y ) i + ( -2y + x) j

at point (3,1)

grad (z) = 4i + j

a)

AB direction = ( B-A) = (4-3)i + (4-1)j=i+3j

unit vector in AB = 1/sqrt(10).(i+3j)

maximum rate of change of z = grad(z).(unit vector in AB direction) = 1/sqrt(10).(4i+j).(i+3j) =7/sqrt(10)=2.21

b)

The direction in which z is increasing fastest is grad(z) at point(3,1) = 4i+j

c)

max rate of change at (3,1) = | grad (z) | = sqrt(16+1) =4.12

d)

P ( 2t , t^2-3)

we have to find at what time p(4,1)

4= 2t --> t= 2

e)

dz/dt = { 2 , 2t } at 4,1 i.e t=2 dz/dt = { 2, 4} rate dz/dt = sqrt { 4+ 16} = sqrt (20)=4.47 inches/sec

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