Sketching Graph: Tips, Hints & Solution

[Gameowner]

Homework Statement

Question attached

Homework Equations

The Attempt at a Solution

I have no idea at all how to start this, I've plotted all 3 separated as x=, y= and z=, which gave me 3 different curves, would I then just join them together?

any hint, tips would be much appareciated

 

[LCKurtz]

x =\sqrt z \sin(\theta),\ y = \frac{\sqrt z} 2 \cos(\theta)

Hint: Multiply the second equation by 2, square both and add and see if that helps.

 

[Gameowner]

x =\sqrt z \sin(\theta),\ y = \frac{\sqrt z} 2 \cos(\theta)

Hint: Multiply the second equation by 2, square both and add and see if that helps.

yes! that has helped a lot, although when it came to plotting the equation, I was not sure about the domain of the graph. But this is what I got

I assume since the question specifies that theta is between 0 and 2pi, the maximum allowable value for x-axis and y-axis is -1/1?

As for the h, which is any number greater than 0, I just substituted an arbitrary point greater than 0.

 

[LCKurtz]

Try this code for a nicer plot,which might help you with your domain. I used h = 10 for an example:

restart;
h := 10;
surface := [sqrt(z)*cos(theta), sqrt(z)*sin(theta), z];
domain := [sqrt(z)*cos(theta), sqrt(z)*sin(theta), 0];
plot3d({domain, surface}, theta = 0 .. 2*Pi, z = 0 .. h, style = patchcontour);