Suppose you are climbing a hill whose shape is given by the equation z=1000-0.00
ID: 2856932 • Letter: S
Question
Suppose you are climbing a hill whose shape is given by the equation z=1000-0.005x^2-0.01y^2. Where x y and z are measured in meters and you are standing at a point with corridinates (100, 120, 806) 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? at what rate? B if you walk northwest will you start to ascend or descend? At what rate? C in which direction is the slope largest? What is the date of ascent in that direction? Suppose whose shape is given by the equation z 1000-0.005x2 -o.01y2, where x, y, and z are measured in meters, and you are standing at you are climbing a hill a point with coordinates (100, 120, 806). 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? vertical meters per horizontal meter (b) If you walk northwest, will you start to ascend or descend? ascend 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 the rate of ascent in that direction? 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.)Explanation / Answer
dz/dx = -0.01x and dz/dy = -0.02*y
del Z = < -0.01*100,-0.02*120> = <-1,-2.4>
a) when he moves south , direction vector =u = <0,-1>
rate = <-1,-2.4>.<0,-1> = 2.4 ---------answer a( this is the rate and because it is +ve , guy ascends)
b)Walking northwest ( = 3/4), the unit directional vector is u = <cos 3/4, sin 3/4>
rate = <-1,-2.4>.<-1/sqrt(2) , 1/sqrt(2) >
= 1/sqrt(2)-2.4/sqrt(2) = -1.4/sqrt(2) = -0.99 -----(this is the rate and because it is -ve , guy descends)
c)
|del Z|= sqrt( 1^2+1.2^2) = 1.562
so slope is maximum in del Z/|delZ| direction
del Z/|del Z| = <-1/1.562 , -1.2/1.562> = <-0.64 , -0.768>
rate = <-1,-1.2>.<-0.64,-0.768> = 1*0.64+1.2*0.768 = 1
direction = arctan(-0.768/-0.64)+pi = 4.017 rad = 230.157 degrees