Thread: Describe an Object Using Spherical Coordinates

1. Describe an Object Using Spherical Coordinates

hey

i'm having a problem with rectangular and spherical coord.

i'm going to describe a body B which limits is the cone z=-(x^2+y^2)^1/2 and the sphere x^2+y^2+z^2=16 , in spherical coordinates. I figured that 0<= rho <=4 but how do i find theta and phi?

thanks!!

2. Have you drawn a diagram they always help!

Now we need to find where they intersect so plug the first equation into the 2nd to get

$x^2+y^2+(-\sqrt{x^2+y^2})^2=16 \iff x^2+y^2=8$

Now this is where the diagram is useful. We only have the bottom half of the cone so the intersection point is below the xy plane.

We also know the distance from the origin to the point in the xy plane where they intersect
$d=\sqrt{8}=2\sqrt{2}$

Now we know the base of the triangle and it hypotenuse $r=4$

So we can use the cosine function to find the angle of depression below the xy plane

$\cos(\alpha)=\frac{2\sqrt{2}}{4}=\frac{\sqrt{2}}{2 } \implies \alpha =\frac{\pi}{4}$

Now you need to figure out how $\alpha$ relates to all of this.

Once again A diagram will be useful.

See if this gets you started and post again if you get stuck.