sphere of raadus2?
a sphere of radius 2 is centered at the origin. A hole of radius1 is drilled though the sphere, with the axis of the hole lying on z axis . Describe the solid region that remains in a) cylindrical coordinates and b) spherical coordinates.
that is express each coordinate variable as a double inequality of the form.
- Anonymous1 decade agoFavorite Answer
♠ draw a picture; it always helps;
sphere’s eqn: x^2 +y^2 +z^2 =R^2;
ball: R<=2; hole: r < 1;
subtracting these solids we get a bead:
x^2+y^2 >= r^2 =1, x^2 +y^2 +z^2 <= R^2 =4;
thence z^2 <= R^2 –r^2, |z| <=√(R^2 –r^2) =√3;
♣ polar coords: x=p*cost, y=p*sint,
where 0 <= t < 2pi;
bead: 0 <= t < 2pi, 1 <= R <= 2, |z| <= √3;
♣ spherical coord: p=R*cosq, z=R*sinq,
where –pi/2 <= q <= pi/2;
bead: 0 <= t < 2pi, 1 <= R <= 2, |q| <= atan(√3) or
–pi/3 <= q <= +pi/3;