Question

over a 24-hour period, the tide in a harbor can be modeled by one period of a sinusoidal function. The tide measures 7ft at midnight, rises to a high of 12ft, falls to a low of 2ft, and then rises to 7ft by the next midnight. What is the equation for the sone function f(x), where x represents time in hours since the beginning of the 24-hour period

Answer 1

The given function will be f(x) = 5 sin(πx/12) + 7

Explanation:

Let the sin function be f(x) = a sin( bx+c) + d

In the given question tide measured at midnight was = 7 ft which rose to a high of 12 ft and fell to a low of 2 ft.

Therefore amplitude a = (12-2)÷2 = 10÷2 = 5

Period b = 2π÷24 = π/12

Horizontal displacement c = 0

Vertical shift d = 7

Now by putting these values in the assumed function.

f(x) = 5 sin(πx/12) + 7

So the right answer is f(x) = 5 sin(πx/12) +7

Answer 2

`y = 5sin((\pi)/(12)x) + 7`

Explanation:

The general equation of a sinusoidal function is:

`y = Asin(wx + c) + s`

Where:

A = amplitude

w = angular velocity =
`(2\pi)/(T)`

T = period =
`(2\pi)/(w)`

c = phase angle

s = vertical displacement.

`A = ((max -min))/(2)`

`A = ((12 -2))/(2)`

`A = 5`

`T = 24\ h`

`w = (2\pi)/(24) = (\pi)/(12)`

`s = (max + min)/(2) = (12 + 2)/(2)`

`s = 7`

`c = 0`

So, the equation is:

`y = 5sin((\pi)/(12)x) + 7`