I'm absolutely stuck with the following problem. I want to find the function that models the following description, assuming the centre of the wheel is the point $(0, 4).$
So, we have a function $f(x) = a\sin(k(x-d))+c.$
From there I got $f(x) = 3\sin(k(x))+4.$ I have no idea what to do now.
To get the function, let's assume that Naill starts at the bottom at $t=0$. In order to get this, we need to shift right by $kd = \frac{\pi}{2}$ (the $\sin$ function normally starts in the middle of it's range). We also know that $90$ seconds is a full period, so $k = \frac{2\pi}{90}$. Therefore, the function is
$$f(x) = 3 \sin\left(\frac{2\pi}{90}\left(x - \frac{90}{4}\right)\right) + 4$$
where $x$ is given in seconds.
You can verify the plot on WolframAlpha.
We don't need the full formula for the domain and range:
The domain is the time on the ride: from $t = 0$ to $t = 10 \cdot 90$ ($10$ revolutions, $90$ seconds each).
The range is the height. Since $-1 \le \sin(x) \le 1$, the range is $(3\cdot(-1)+4, 3\cdot(1)+4) = (1, 7)$