Given the population data of lily pads over time, we want to model this data using an exponential function. The general form of an exponential function is:
[tex]\[ y = a \cdot b^x \][/tex]
where:
- [tex]\( y \)[/tex] is the population at time [tex]\( x \)[/tex],
- [tex]\( a \)[/tex] is the initial population (at [tex]\( x = 0 \)[/tex]),
- [tex]\( b \)[/tex] is the growth factor.
The provided data points are:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time (x)} & \text{Population (y)} \\
\hline
0 & 4 \\
\hline
5 & 7 \\
\hline
10 & 10 \\
\hline
15 & 15 \\
\hline
20 & 33 \\
\hline
25 & 51 \\
\hline
30 & 79 \\
\hline
\end{array}
\][/tex]
Using an exponential regression analysis of this data, we obtain values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] as follows:
- [tex]\( a = 4.269 \)[/tex]
- [tex]\( b = 1.103 \)[/tex]
These values are rounded to the nearest thousandth.
Therefore, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] for the exponential function modeling the water lily population growth are:
[tex]\[ a = 4.269 \][/tex]
[tex]\[ b = 1.103 \][/tex]
