To model the given data using an exponential function, we first need to recognize the form of an exponential function, which is typically written as:
[tex]\[ y = a b^x \][/tex]
Our goal is to determine the values of \(a\) and \(b\) that best fit our data. Given our data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & 1 \\
\hline
10 & 10 \\
\hline
15 & 15 \\
\hline
20 & 33 \\
\hline
25 & 51 \\
\hline
30 & 79 \\
\hline
\end{array}
\][/tex]
We can use a method like least-squares regression to fit an exponential curve to this data. The regression tool will compute the values of \(a\) and \(b\) that minimize the sum of the squared differences between the observed values (y-data) and the values predicted by the exponential function.
After performing the regression analysis, we determine the best-fitting values for \(a\) and \(b\).
The values, rounded to the nearest thousandth, are:
[tex]\[ a = 3.843 \][/tex]
[tex]\[ b = 1.107 \][/tex]
Therefore, the exponential model that fits the water lily population growth is:
[tex]\[ y = 3.843 \cdot 1.107^x \][/tex]
Thus, the values are [tex]\(a = 3.843\)[/tex] and [tex]\(b = 1.107\)[/tex].
