The following diagram represents a large Ferris wheel, with a diameter of 100 meters.
Let P be a point on the wheel. The wheel starts with P at the lowest point, at ground level. The wheel rotates at a constant rate, in an
anticlockwise (counter-clockwise) direction. One revolution takes 20 minutes.
Given that h can be expressed in the form $h(t) = a\cos bt + c$, find $a$, $b$ and $c$.
I understand that $b$ is $\frac{\pi}{10}$. The answer key gives that $a = -50$ and $c = 50$, but how do I show this?
Thanks
Best Answer
Since the period is $20$ minutes, $$20~\text{min}= \frac{2\pi}{b} \implies b = \frac{2\pi}{20~\text{min}} = \frac{\pi}{10}~\frac{1}{\text{min}}$$ so you worked out the frequency correctly.
Since point $P$ is initially at the bottom, $h(t) = 0~\text{m}$ when $t = 0$. Therefore, $$h(0) = a\cos(0) + c = a + c = 0~\text{m}$$
Since point $P$ is at the top halfway through one revolution and a revolution lasts $20$ minutes, point $P$ reaches the top after ten minutes. Since the ferris wheel is $100~\text{m}$ tall, $h(10) = 100~\text{m}$. Thus, $$h(10) = a\cos(\pi) + c = -a + c = 100~\text{m}$$ That leaves you with a system of two linear equations in two variables to solve.