Suppose that you are climbing a hill whose shape is given by theequation z = 1000 -0.01x2 -0.02y2 and you are standing at a pointwith coordinates (60, 100, 764).(a) In which direction should you proceed initially inorder to reach the top of the hill fastest? (Enter as a vector ofmagnitude 1.)&lt; ___, ___&gt;(b) If you climb in that direction, at what angle above thehorizontal (in degrees) will you be climbing initially?_____

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Suppose that you are climbing a hill whose shape is given by theequation z = 1000 -0.01x2 -0.02y2 and you are standing at a pointwith coordinates (60, 100, 764).(a) In which direction should you proceed initially inorder to reach the top of the hill fastest? (Enter as a vector ofmagnitude 1.)&amp;lt; ___, ___&amp;gt;(b) If you climb in that direction, at what angle above thehorizontal (in degrees) will you be climbing initially?_____

Pegoxv · Accepted Answer

For this question you first need to find the partialderivatives. After which we create a gradient vector from the pointand the partial derivatives. The gradient Vector is alwaysperpendicular to the function at the point it originates. So tostart this problem, we first need to find the partialderivatives.Taking the derivative with respect to X only we get:fx = -.02xTaking the derivative with respect to Y only we get:fy = -.04yNow to get the gradient vector we follow the formula for computingthe gradient vector from partial derivatives.fx i + fy j= grad fso-.02xi -.04yj = grad f which =-.02(60)i-.04(100)j simplified, grad f =-1.2i - 4jnow to turn this into a unit vector we need to divide by themagnitude of the vector. This will result in the vectors magnitudebeing = 1. The magnitude is calculated as follows:      −    1.2    2  +  (  −  4  )  2  )  ;  s  i  m  p  l  i  f  i  e  d  =      17.44  so your unit vector is:   −  1.2      17.44    i  −  4      17.44    jthe second half should be the calculated angle between the horizonand the y=x plane and the gradient we found.

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