Sinusoidal modeling

Building the model is a matching exercise, and the matches never change:

1. Amplitude = radius (or half the total swing). A wheel of diameter 40 m has amplitude 20, because a rider strays at most one radius from the hub's height.
2. Midline = center height: where the hub sits, or the average of the context's max and min. A wheel whose lowest point is a 2 m platform with radius 20 has its hub, and midline, at 22 m.
3. Period = time for one full cycle: one revolution, one full tide cycle, one year.
4. Starting position picks the form (next section).

Then assemble $h(t) = a\,\sin\!\big(\tfrac{2\pi}{\text{period}}\,t\big) + k$ or the cosine version. Note the b slot gets $2\pi/\text{period}$, not the period itself.

A ferris wheel has diameter 40 m, its boarding platform is 2 m off the ground at the wheel's lowest point, and one revolution takes 6 minutes. Extract: radius 20, so amplitude 20. Lowest point 2 m, so the hub is at 2 + 20 = 22 m: midline 22. The rider's height runs from 2 m up to 42 m, which checks against midline ± amplitude. Period 6, so b = 2π/6 = π/3.

The two numbers most often stolen by mistakes: the diameter 40 (which is twice the amplitude) and the platform height 2 (which is the minimum, not the midline). Every parameter should be sanity-checked against the physical extremes: does midline + amplitude equal the real maximum? Does midline − amplitude equal the real minimum?

The starting position at t = 0 decides the cleanest form:

1. Starts at the maximum: use $a\cos(bt) + k$.
2. Starts at the minimum: use $-a\cos(bt) + k$.
3. Starts at the midline moving up: use $a\sin(bt) + k$.
4. Starts at the midline moving down: use $-a\sin(bt) + k$.

Sine and cosine are not interchangeable without a shift: sine starts mid-swing, cosine starts at the crest. Using sine for a rider who boards at the bottom silently claims they boarded at hub height. Any start position not on this list is handled with a phase shift, but AP contexts overwhelmingly start at one of the four landmarks.

Once built, the model answers questions two ways. For "what is the height at t = ...?", substitute and evaluate. For "when does it first reach ...?", think in quarter cycles before reaching for algebra: a rider boarding at the bottom reaches hub height after a quarter turn, the top after half a turn, and is back at the bottom after a full turn.

Quarter-cycle reasoning also catches substitution errors. If the period is 60 s and the ride starts at the midline going up, the peak arrives at t = 15, one quarter of 60, and any algebra that says otherwise has slipped. Keep the period, not b, in your head for timing questions, and convert only when writing the formula.