To determine the period of the function that models the height of the tide ([tex]$h(t)$[/tex]) over time ([tex]$t$[/tex]), let's analyze the given data in the table:
[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]
First, we should observe how the values of [tex]$h(t)$[/tex] change as [tex]$t$[/tex] increases. The behavior over time should help us identify if and when the function repeats.
- At [tex]$t = 0$[/tex], [tex]$h(t) = 5$[/tex].
- At [tex]$t = 1$[/tex], [tex]$h(t) = 4.41$[/tex].
- At [tex]$t = 2$[/tex], [tex]$h(t) = 3$[/tex].
- At [tex]$t = 3$[/tex], [tex]$h(t) = 1.59$[/tex].
- At [tex]$t = 4$[/tex], [tex]$h(t) = 1$[/tex].
- At [tex]$t = 5$[/tex], [tex]$h(t) = 1.59$[/tex].
By examining the values of [tex]$h(t)$[/tex], we notice the following:
- The height of the tide at [tex]$t = 5$[/tex] is again [tex]$1.59$[/tex], which is the same as at [tex]$t = 3$[/tex]. This suggests that the function may start repeating its values.
- If we observe further, the initial value of [tex]$h(t)$[/tex] at [tex]$t = 0$[/tex] is [tex]$5$[/tex], and the function values decrease and then again tend to start repeating after reaching [tex]$t = 5$[/tex].
This repetitive pattern indicates that the period of the function is the length of time it takes for the function to repeat itself. In this case, it is evident from the data that the period is 5 hours.
Therefore, the correct answer is:
A. 5 hours
