Please help me with this calculus question!

Question

phi · Answer

the first equation is a sphere centered at (0,0,1/2) with radius 1/2.
to see this, complete the square on the z variable
$$ x^2 + y^2 + z^2 =z \ x^2 + y^2 + z^2-z +(1/4) = 1/4 \
x^2+ y^2 + (z-\frac{1}{2})^2 = \frac{1}{4} $$

in cylindrical coords,
$$ r^2 + \left(z-\frac{1}{2}\right)^2 = \frac{1}{4} \
r= \sqrt{ \frac{1}{4} -\left(z-\frac{1}{2}\right)^2 }
 $$

phi · Answer

the second equation is a cone with its apex at the origin
$$ z= \frac{1}{\sqrt{3} }\sqrt{x^2+y^2} $$
in cylindrical coords
$$ z =  \frac{r}{\sqrt{3} } \ r = \sqrt{3} \ z
$$

phi · Answer

the composite shape is similar to an ice-cream cone:
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