Tides rise and fall, daylight hours change through the year, and a capsule on a big wheel goes round and round. Anything that repeats at a steady rate can be modelled by y=asin bt+c or y=acos bt+c, where t is time. **What each feature means in a model** **Graph feature** | **Real-world meaning** Midline y=c | The average (central) level, e.g. mean water depth or the height of the wheel's axle. Amplitude a | The largest swing away from the average, e.g. tidal range, 2 or the wheel's radius. Period 2pib or 360 b | The time for one complete cycle, e.g. one rotation or one full year. Maximum c+a, minimum c-a | The highest and lowest values the quantity ever reaches. **A tide model: h=3sin pi t6+5, 0 <= t <= 24 (h in metres, t in hours)** Mean depth 5 m (midline), amplitude 3 m, period 2pipi/6=12 hours — two full tides in the 24-hour window. High tide 8 m at t=3; low tide 2 m at t=9. - Identify a, b and c, then state amplitude, period 2pib and midline in the units of the question. - To find the value at a given time, substitute t into the model. - Greatest and least values are c+a and c-a; they occur when the sine or cosine factor equals 1 or -1. - The quantity equals the average level c when the sine or cosine factor equals 0; sketch the curve and read every time in the stated interval. If the model contains pi, the angle is in radians — set the calculator to radian mode, and check that every time you give lies inside the stated interval.
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