Suppose the high tide in seattle occurs at 1: 00 a. m. and 1: 00 p. m. at which time the water is 16 feet above the height of low tide. low tides occur 6 hours after high tides. suppose there are two high tides and two low tides every day and the height of the tide varies sinusoidally.

(a) find a formula for the function y = h(t) that computes the height of the tide above low tide at time t, where t indicates the number of hours after midnight. (in other words, y = 0 corresponds to low tide.)

h(t) =

(b) what is the tide height at 11: 00 a. m.?

ft

(a)

(b) Height at 11:00 a.m. = 12 feet

Explanation:

Sinusoidal Function

It refers to a mathematical curve with a smooth and periodic oscillation. Its name comes from the sine function but it can be a cosine function too. They only differ by the phase angle of 90 degrees or radians.

The sine function is characterized by

The minimum value is -1 when the argument is radians or 270 degrees

The maximum value is 1 when the argument is or 90 degrees

It completes a full cycle in radians (or 360 degrees)

It's zero at 0 and

It repeats itself along infinite cycles with the same characteristics

We need to model the height of the tide above low tide at time t, t expressed in hours from midnight

We know the following data

At 1:00 and 13:00, the tide is high at 16 feet above the low tide, assumed to be 0 m

At 7:00 and 19:00, the tide is low at 0 m.

(a) The general sine function is expressed as

Where A is the amplitude, w is the angular frequency, is the phase, B is the vertical shift

We know that at t=1, the sine must be at max (value =1) and at t=7 it must be at min (value=-1). Replacing in [1]

We get these equations

Using the same data as before, when t=1 the argument of the sine must be , so it reaches it max, and when t=7, the argument must be . Then:

Solving, we get

The function is complete now

Factoring

(b) We need to find h when t=11