sinusoidal functions. fill in the blanks given the information.

Question

On a certain day the tide in Boston Harbor reaches high tide (10 ft) twice at midnight and at 12:15 PM . if low tide is at a height of 0 feet and T(h) is the height of the tide at time h hours (with hour 0 occurring at midnight on the day in question then:

1. when is the next instance of high tide
2. what is the period of T(h)
3. what is the fourth time that low tide occurs
4. find a formula for T(h)

Answer 1

1.

The next instance of high tide is 30 minutes after midnight the next day.

The reason is that (according to the given information)

high tide occur every 12 hours, 15 minutes.

2.

The period P is the time between two high tides, which is 12 hours, 15 minutes,

i.e. P = 12.25 hours

3.

Low tide occurs with the same period of 12 hours and 15 minutes,

and halfway between high tides.

Therefore it is at

6 o'clock am, 7 minutes, 30 seconds,

6 o'clock pm, 22 minutes, 30 seconds,

6 o'clock am, 37 minutes, 30 seconds,

6 o'clock pm, 52 minutes, 30 seconds.

4.

Being periodic, the formula is of the form

T(h) = C + Asin(2πh/P + φ)

where

C is the average tide height,

A is the amplitude,

P is the period, already given as P = 12.25,

φ is the phase.

The average between 0ft and 10ft is 5ft, therefore

C = 5

The amplitude is the difference between low tide and C.

Since low tide is 0, we get

A = 5

The phase φ is determined from given value of T(0).

Assuming that time starts at that midnight of that "certain day"

when the tide is 10 ft, we get the equation

10 = 5 + 5sin(2π×0/12.25 + φ)

5 = 5sin(φ)

1 = sin(φ)

φ = π/2

So the formula is

T(h) = 5 + 5sin(2πh/12.25 + π/2)