Over a 24-hour period, the tide in a harbor can be modeled by one period of a sinusoidal function. The tide measures 4.35 ft at midnight, rises to a high of 8.3 ft, falls to a low of 0.4 ft, and then rises to 4.35 ft by the next midnight. What is the equation for the sine function f (x), where x represents time in hours since the beginning of the 24-hour period, that models the situation?

f (x) =

f (x) = 4.35 + 3.95·sin (πx/12)

Step-by-step explanation:

For problems of this sort, a sine function is used that is of the form ...

f (x) = A + Bsin (2πx/P)

where A is the average or middle value of the oscillation, B is the one-sided amplitude, P is the period in the same units as x.

It is rare that a tide function has a period (P) of 24 hours, but we'll use that value since the problem statement requires it. The value of A is the middle value of the oscillation, 4.35 ft in this problem. The value of B is the amplitude, given as 8.3 ft - 4.35 ft = 3.95 ft. Putting these values into the form gives ...

f (x) = 4.35 + 3.95·sin (2πx/24)

The argument of the sine function can be simplified to πx/12, as in the Answer, above.