Question

find an equation in cylindrical coordinates for the surface represented by the rectangular equation. y^{2}=80 - z^{2}

Explanation:

Step1: Recall the relationship in cylindrical coordinates

In cylindrical coordinates, $y = r\sin\theta$.

Step2: Substitute $y$ into the given equation

Substitute $y = r\sin\theta$ into $y^{2}=80 - z^{2}$. We get $(r\sin\theta)^{2}=80 - z^{2}$, which simplifies to $r^{2}\sin^{2}\theta+z^{2}=80$.

Answer:

$r^{2}\sin^{2}\theta + z^{2}=80$