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.) b) If you climb in that direction, at what angle above thehorizontal (in degrees) will you be climbing initially?

Jaydin Harvey

Jaydin Harvey

Answered question

2022-09-03

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.)
< ___, ___>
(b) If you climb in that direction, at what angle above thehorizontal (in degrees) will you be climbing initially?
_____

Answer & Explanation

Pegoxv

Pegoxv

Beginner2022-09-04Added 7 answers

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 = -.02x
Taking the derivative with respect to Y only we get:fy = -.04y
Now to get the gradient vector we follow the formula for computingthe gradient vector from partial derivatives.
fx i + fy j= grad f
so
-.02xi -.04yj = grad f which =-.02(60)i-.04(100)j simplified, grad f =-1.2i - 4j
now 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 j
the second half should be the calculated angle between the horizonand the y=x plane and the gradient we found.

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