Tides

Part of the Graphs of Trig Functions section of Coddy's Trigonometry journey. Lesson 42 of 51.

In many harbours the water rises and falls twice a day. Say high tide is 10 m at midnight, low tide is 4 m six hours later, and the pattern repeats every 12 hours (real tides take about 12 hours 25 minutes). Time t is in hours, so the angle has to turn 360° in 12 hours: 30° per hour.

d = 3cos(30°×t) + 7

The midline is (10 + 4)/2 = 7, the amplitude (10 − 4)/2 = 3, and the depth starts at its top at t = 0, so it is a cosine with a positive a. The number in front of t is degrees per hour: 360° divided by the period in hours.

Challenge

On a coast with one high and one low tide a day, high water is 5 m at t = 0 and low water is 1 m twelve hours later. Take the pattern to repeat every 24 hours. Write the depth as d = a×cos(b×t) + k, with b in degrees per hour written like 30°t. Press Check.