find an equation in rectangular coordinates for the surface represented by the cylindrical equation. r = 4…

Explanation:

Step1: Recall the relationship between cylindrical and rectangular coordinates

We know that $x = r\cos\theta$, $y = r\sin\theta$ and $r^{2}=x^{2}+y^{2}$. Given $r = 4\sin\theta$, multiply both sides by $r$. $r^{2}=4r\sin\theta$

Step2: Substitute rectangular - coordinate expressions

Since $r^{2}=x^{2}+y^{2}$ and $y = r\sin\theta$, substituting these into the equation $r^{2}=4r\sin\theta$ gives $x^{2}+y^{2}=4y$.

Step3: Rearrange the equation

Move all terms to one side: $x^{2}+y^{2}-4y = 0$. Then complete the square for the $y$ - terms. We have $x^{2}+y^{2}-4y+4 = 4$, which can be written as $x^{2}+(y - 2)^{2}=4$.

Answer:

$x^{2}+(y - 2)^{2}=4$