Thames River depth When a river flows into an ocean, the depth of the river varies near its mouth as a result of tides. Information about this change in depth is critical for safety. The following table gives the depth D (in feet) of the Thames River in London for a 24-hour period.
Time | D | Time | D | Time | D |
12 A.M. | 27.1 | 8 A.M. | 20.0 | 4 P.M. | 34.0 |
1 A.M. | 30.1 | 9 A.M. | 18.0 | 5 P.M. | 32.4 |
2 A.M. | 33.0 | 10 A.M. | 18.3 | 6 P.M. | 29.1 |
3 A.M. | 34.3 | 11 A.M. | 20.6 | 7 P.M. | 25.2 |
4 A.M. | 33.7 | 12 P.M. | 24.2 | 8 P.M. | 21.9 |
5 A.M. | 31.1 | 1 P.M. | 28.1 | 9 P.M. | 19.6 |
6 A.M. | 27.1 | 2 P.M. | 31.7 | 10 P.M. | 18.6 |
7 A.M. | 23.2 | 3 P.M. | 33.7 | 11 P.M. | 19.6 |
(a) Plot the data, with time on the horizontal axis and depth on the vertical axis. Let t = 0 correspond to 12:00 A.M.
(b) Determine a function D(t)= a sin (bt + c)+ d, where D(t) represents the depth of the water in the harbor at time t. Graph the function D with the data. (Hint: To determine b, find the time between maximum depths.)
(c) If a ship requires at least 24 feet of water to navigate the Thames safely, graphically determine the time inter-val(s) when navigation is not safe.
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