To determine the period of the function that models the height of the tide at Sunny Beach, let's analyze the given data and observations:
### Data Analysis
We have height [tex]\( h(t) \)[/tex] corresponding to various times [tex]\( t \)[/tex] such as:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
t & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
h(t) & 5 & 4.41 & 3 & 1.59 & 1 & 1.59 \\
\hline
\end{array}
\][/tex]
### Key Observations
1. Highest Tide Occurrence: According to the problem, the highest tide occurs at 12:00 p.m., which corresponds to [tex]\( t=0 \)[/tex].
2. Symmetry in Data: The data suggests symmetry about [tex]\( t = 2.5 \)[/tex] hours, where the highest and lowest points indicate a cyclical pattern:
- At [tex]\( t = 0 \)[/tex] hours, the height is [tex]\( h(0) = 5 \)[/tex] (a peak).
- At [tex]\( t = 5 \)[/tex] hours, the height returns to around [tex]\( h(5) = 1.59 \)[/tex], close to the starting value but still generally modelled as completing one cycle.
### Understanding the Period
A trigonometric function, often sin or cos, which models periodic behavior, completes one full cycle when it returns to the starting position. Given the data:
- The values show that heightened starts high at [tex]\( t = 0 \)[/tex].
- They appear to have returned to a similar (peak) structure around [tex]\( t = 5 \)[/tex], suggesting halfway through another cycle or a cyclical return.
### Conclusion
From peak to peak (from [tex]\( t=0 \)[/tex] to [tex]\( t=5 \)[/tex]), it completes one cycle. Therefore, the complete period to repeat this range is twice this interval, indicating a full cycle every 12 hours.
Thus, the period of the function that can model the height of the tide is
### 12 hours.
The correct answer is C. 12 hours.
