The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 a.m. and 3:30 p.m., with a depth of 3 point 2 5 meters, while high tides occur at 7:45 a.m. and 11:15 p.m., with a depth of 8 point 7 5 meters. Which of the following equations models d , the depth of the water in meters, as a function of time, t , in hours? Let t is equal to 0 be 12:00 a.m.
Answer
A
d is equal to negative 2 point 7 5 cosine open paren 4 pi over 29 t close paren plus 6
B
d is equal to negative 2 point 7 5 cosine open paren 4 pi over 31 t close paren plus 6
C
d is equal to negative 3 point 7 5 cosine open paren 4 pi over 31 t close paren plus 5
D
d is equal to negative 3 point 7 5 cosine open paren 4 pi over 29 t close paren plus 5
Explanation
The midline is the average of the high and low depths: (8.75 + 3.25) / 2 = 6 meters. The amplitude is half their difference: (8.75 − 3.25) / 2 = 2.75 meters. Consecutive low tides are 15.5 hours apart, so the period is 15.5 hours and the cosine factor is 2π / 15.5 = 4π / 31. Since the depth is at a low tide when t = 0, the cosine term needs a negative amplitude. These values give choice A: d = −2.75 cos(4πt / 31) + 6.
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