Answer:
[tex]f(t)=4\cdot \cos\left(\dfrac{\pi}{6}\cdot t\right) + 5[/tex]
Step-by-step explanation:
As it is a fluctuating scenario, where the value repeats itself after some time, it must be represented by periodic function.
As the value at the beginning is a non zero value, we should use a Cosine function.
The general cosine function is,
[tex]f(t) = a\cdot \cos\left(\dfrac{2\pi}{p}\cdot t\right) + b[/tex]
where,
a = amplitude of motion,
p = period
,
b = vertical displacement.
The water level at high tide was 9 feet and 1 foot at low tide. So,
[tex]a=\dfrac{\text{Max height - Min height}}{2}=\dfrac{9-1}{2}=\dfrac{8}{2}=4[/tex]
[tex]b=\dfrac{\text{Max height + Min height}}{2}=\dfrac{9+1}{2}=\dfrac{10}{2}=5[/tex]
The next high tide is exactly 12 hours later. So period or p = 12
Putting all the values,
[tex]f(t) = 4\cdot \cos\left(\dfrac{2\pi}{12}\cdot t\right) + 5=4\cdot \cos\left(\dfrac{\pi}{6}\cdot t\right) + 5[/tex]
