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How steep is this hill

  1. Jul 23, 2010 #1
    1. The problem statement, all variables and given/known data

    The height of a certain hill is given by h(x,y)=10(3xy-4x^2-2y^2-11x+17y+9)

    (a) How steep is the slope in the x and in the y direction (in meters per kilometer) at the point 1 km North and 1 km
    East of Dandenong?
    (b) In what direction is the slope the steepest at that point and how steep is the slope in that
    direction?

    2. Relevant equations


    b)grad(1,1)=-160i-160j
    |grad(1,1)|=226km/km

    now for part a what does it mean when they ask how steep is the slope in the x and y direction?
    grad(1,1)=-160i-160j

    are they asking for the co-ordinates? or the magnitude of each seperate component?
    anyway this has me a bit confused.
    3. The attempt at a solution
     
  2. jcsd
  3. Jul 23, 2010 #2
    Re: gradF

    Presumably, you are also told where Dandenong is in relation to the hill?
     
  4. Jul 23, 2010 #3
    Re: gradF

    The height of a certain hill is given by h(x,y)=10(3xy-4x^2-2y^2-11x+17y+9)
    where y is the distance north of dandenong and x is the distance east of dandenong.
     
  5. Jul 23, 2010 #4
    Re: gradF

    OK, the question is asking about the gradient in the x direction and the gradient in the y direction at the point x=1, y=1.
    This is clearly an exercise in partial differentials.

    I have no idea what your 'relevant equations' are - they don't look relevant at all (EDIT - in fact they look like the answer values).
    To obtain the gradient in x, you simply differentiate with respect to x, treating y as a constant. Similarly for y.
    Then you plug in the values for the point (1,1) to those differentials. The result is the gradient (slope) in the direction chosen.

    Then having obtained those values, you are asked to find the direction of maximum slope at that point and give a value to that slope.
     
    Last edited: Jul 23, 2010
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