Classical Dynamics prob, please.

[ChronicQuantumAddict]

The question is as follows: The height of a hill (meters) is given by [z=(2xy)-(3x^2)-(4y^2)-(18x)+(28y)+12], where x is the distance east, y is the distance north of the origin. a). where is the top of the hil (x,y,z) and how high is it (z=?)? b). How steep is the hill at x=y=1, that is, what is the angle between a vector perpendicular to the hill and the z axis? c). In which compass direction is the slope at x=y=1 the steepest?

okay, i know part a)., which is to get the derivative of dz/dx and dz/dy to get your x and y maximum values, then plug them back into the equation for z, giving you the location and the height of the hill at the steepest point.

The part i am having trouble with is b). how am i supposed to draw a vector that is perpendicular to the hill and calculate hte angle it makes with the z-axis? i thought of plugging in the x and y values of x=y=1 into the z equation, and this would give you z at that point, then create a vector normal to that point, but how do u find the angle (theta) made with the z-axis.

I also figured out part c). which is in a south east direction.

Please help, homework is due soon, thank you.

 

[HallsofIvy]

Think of f(x,y,z)= constant (here that would be (2xy)-(3x^2)-(4y^2)-(18x)+(28y)+12- z= 0) as "level surface" for f. Then the gradient of f (the vector whose components are f_x, f_y, f_z) is perpendicular to the surface (evaluated at (1,1,37)). You can find the angle it makes with the z-axis by taking its dot product with (0, 0, 1).

 

[ChronicQuantumAddict]

thank u

 

[SEMonChron]

this is a major bump, but it is similar to a problem i have. would someone else explain this?

where did (1,1,37) come from?

i think it is easy, but i am just drawing a huge blank...

and then the gradient of f is (fx, fy, fz) where fx is your dz/dx you needed to find to solve part a? and fy is your dz/dy, then fz would be 1? or am i totally wrong.

thanks for any help! I am new to this site, it is a great help

 

[cashmerelc]

this is a major bump, but it is similar to a problem i have. would someone else explain this?

where did (1,1,37) come from?

i think it is easy, but i am just drawing a huge blank...

and then the gradient of f is (fx, fy, fz) where fx is your dz/dx you needed to find to solve part a? and fy is your dz/dy, then fz would be 1? or am i totally wrong.

thanks for any help! I am new to this site, it is a great help

In this case, you don't have to find fz. Z itself is f(x,y). The value for z comes from plugging in the values for x and y into the equation which gives you z.