Answer
The function is neither linear nor exponential.
Work Step by Step
30)
Given
$$
\begin{array}{c|c|c|c}
\hline t & 5 & 12 & 16 \\
\hline S(t) & 4.35 & 6.72 & 10.02 \\
\hline
\end{array}
$$
A function is linear if the rate of change is a constant.
$$
\begin{aligned}
& \frac{6.72-4.35}{12-5}=0.339 \\
& \frac{10.02-6.72}{16-12}=0.825 .
\end{aligned}
$$
The function is not linear. We now check the ratios to see if the function is exponential.
$$
\begin{aligned}
\frac{S(12)}{S(5)}=\frac{a b^{12}}{a b^5} & =\frac{6.72}{4.35} \\
b^7 & =\frac{6.72}{4.35} \\
b & \approx 1.064
\end{aligned}
$$
$$
\begin{aligned}
\frac{S(16)}{S(12)}=\frac{a b^{16}}{a b^{12}} & =\frac{10.02}{6.72} \\
b^4 & =\frac{10.02}{6.72} \\
b & \approx 1.105 .
\end{aligned}
$$
The function is neither linear nor exponential.
