Question

Over a 24-hour period, the tide in a harbor can be modeled by one period of a sinusoidal function. The tide measures 5 ft at midnight, rises to a high of 9.5 ft, falls to a low of 0.5 ft, and then rises to 5 ft by the next midnight.

Give the value of each given that x represents time in hours since the beginning of the 24 hour period that models the situation.

Answer 1

Explanation:

The tide over a 24-hour period can be modeled by a sinusoidal function of the form:

f(x) = A * sin(Bx) + C

where A is the amplitude, B is the frequency, and C is the vertical shift.

To determine the values of A, B, and C, we need three points. Let's use the following points:

(0, 5): This is the value of the tide at midnight, 5 ft.

(12, 9.5): This is the value of the tide at its high, 9.5 ft.

(6, 0.5): This is the value of the tide at its low, 0.5 ft.

Using the first point (0, 5), we can calculate the vertical shift, C:

C = 5

Using the second point (12, 9.5), we can calculate the amplitude, A:

A = (9.5 - 5) / 2 = 2.25

Using the third point (6, 0.5), we can calculate the frequency, B:

0.5 = 2.25 * sin(B * 6) + 5

4.5 = 2.25 * sin(B * 6)

2 = sin(B * 6)

B * 6 = sin^-1(2)

B * 6 = pi / 2

B = pi / (2 * 6) = pi / 12

So, the final equation is:

f(x) = 2.25 * sin(pi * x / 12) + 5

This represents one period of the sinusoidal function that models the tide over a 24-hour period, with time x measured in hours since midnight.