From one hour to the next, how does an exponential growth factor change the population: do you add a fixed amount or multiply by a fixed number?

Compute $\dfrac{P_1}{P_0}$, $\dfrac{P_2}{P_1}$, and so on. These ratios should all be about the same; which answer choice is closest to that common value?

In Desmos, type the expressions 1.8/1, 3.2/1.8, 5.8/3.2, and 10.5/5.8 on separate lines. Look at the decimal values Desmos gives for each ratio.

Compare the decimal values of those ratios to the four answer options (1.5, 1.6, 1.8, 2.0). Identify which option is closest to all of the ratios shown by Desmos; that option is the hourly growth factor.

The model is $P = A r^t$, where $P$ is the population (in thousands), $A$ is the initial population when $t=0$, and $r$ is the hourly growth factor (the multiplier each hour). At $t=0$, the table shows $P=1.0$, so $A=1.0$ and the model becomes $P = r^t$.

For exponential models, the ratio of the population at one hour to the previous hour is the growth factor $r$.

So, if the data fit an exponential model, the ratios $\frac{P_1}{P_0}$, $\frac{P_2}{P_1}$, $\frac{P_3}{P_2}$, and $\frac{P_4}{P_3}$ should all be about the same, and that common value is $r$.

Use the given values (in thousands):

• From $t=0$ to $t=1$: $\dfrac{P_1}{P_0} = \dfrac{1.8}{1.0} = 1.8$.
• From $t=1$ to $t=2$: $\dfrac{P_2}{P_1} = \dfrac{3.2}{1.8} \approx 1.78$.
• From $t=2$ to $t=3$: $\dfrac{5.8}{3.2} \approx 1.81$.
• From $t=3$ to $t=4$: $\dfrac{10.5}{5.8} \approx 1.81$.

These ratios are all close to each other, so they are good estimates of $r$.

All of the ratios you computed are around $1.8$, and this is the only answer choice that matches those values closely. Therefore, the hourly growth factor $r$ is approximately $1.8$, so the correct choice is C.