Answer
$x^2 + (y - 290)^2 = 67, 600$
Work Step by Step
The minimum height of the ferris wheel is equal to:
\begin{align*}
\text{Maximum height of the wheel} - \text{diameter of the wheel} &= 550 - 520\\
&= 30 \text{ feet}
\end{align*}
The points $(0, 30)$ and $(0, 550)$ lie on the circle and they are the end points of the vertical diameter of the ferris wheel.
Since the center of a circle is the midpoint of a diameter, then the center is at:
$\left(\dfrac{0+0}{2}, \dfrac{30+550}{2}\right) = (0, 290)$
The radius $r$ is half of the diameter so $r=\frac{520}{2}=260$.
Thus, with $r=260$ and center at $(0, 290)$, the standard equation of the ferris wheel is
\begin{align*}
(x - 0)^2 + (y - 290)^2 &= 260^2\\
x^2 + (y - 290)^2 &= 67, 600
\end{align*}
