Answer:
(a)
[tex]\displaystyle h(t)=y(t)=8\left [ 1+sin\left ( \frac{\pi}{6}t+\frac{\pi}{3} \right ) \right ][/tex]
(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 [tex]\pi/2[/tex] radians.
The sine function is characterized by
The minimum value is -1 when the argument is [tex]3\pi/2[/tex] radians or 270 degrees
The maximum value is 1 when the argument is [tex]\pi/2[/tex] or 90 degrees
It completes a full cycle in [tex]2\pi[/tex] radians (or 360 degrees)
It's zero at 0 and [tex]\pi[/tex]
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
[tex]y(t)=Asin(wt+\phi)+B\ \ .........[1][/tex]
Where A is the amplitude, w is the angular frequency, [tex]\phi[/tex] 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]
[tex]y(1)=A(1)+B=A+B=16[/tex]
[tex]y(7)=A(-1)+B=A-B=0[/tex]
We get these equations
[tex]A+B=16[/tex]
[tex]A=B[/tex]
[tex]Solving, A=8,\ B=8[/tex]
Using the same data as before, when t=1 the argument of the sine must be [tex]\pi/2[/tex], so it reaches it max, and when t=7, the argument must be [tex]3\pi/2[/tex]. Then:
[tex]w+\phi=\pi/2[/tex]
[tex]7w+\phi=3\pi/2[/tex]
Solving, we get
[tex]w=\pi/6[/tex]
[tex]\phi=\pi/3[/tex]
The function is complete now
[tex]\displaystyle y(t)=8sin\left ( \frac{\pi}{6}t+\frac{\pi}{3} \right )+8[/tex]
Factoring
[tex]\displaystyle h(t)=y(t)=8\left [ 1+sin\left ( \frac{\pi}{6}t+\frac{\pi}{3} \right ) \right ][/tex]
(b) We need to find h when t=11
[tex]\displaystyle h(11)=8\left [ 1+sin\left ( \frac{\pi}{6}11+\frac{\pi}{3} \right ) \right ][/tex]
[tex]\displaystyle h(11)=8\left [ 1+sin\left ( \frac{13\pi}{6} \right ) \right ][/tex]
[tex]h(11)=8\left (1+0.5 \right)[/tex]
[tex]h(11)=12\ feet[/tex]
