Ask yourself: which type of model would have a constant increase each hour? Which would have increases that change in a regular pattern? Does the table show either of those behaviors?

Divide each culture size by the previous one (for example, $1.6/1.0$, $2.5/1.6$, and so on). Are these ratios close to each other, or do they vary a lot? What kind of growth is suggested when values multiply by about the same factor each step?

Create a table, put the time values $0,1,2,3,4,5$ in the first column (as $x_1$), and the culture sizes $1.0,1.6,2.5,3.9,6.2,9.8$ in the second column (as $y_1$). This lets you see the scatterplot.

Type y1 ~ m x1 + b to perform a linear regression. Look at how well the line follows the plotted points. If the points clearly bend away from the line and the errors grow, a linear model is not ideal.

Type y1 ~ a x1^2 + b x1 + c for a quadratic regression and y1 ~ a + b ln(x1) for a logarithmic regression. Compare how closely each curve follows the plotted points across the entire range of $t$ values.

Type y1 ~ a b^{x1} to fit a model where values grow by a roughly constant factor each step. Compare the curves from all your regressions and see which one hugs the data points most closely over all times; the type of that best-fitting model is the answer choice you should select.

Look at how much the culture size increases each hour.

From the table:

• From $t=0$ to $t=1$: $1.6-1.0 = 0.6$
• From $t=1$ to $t=2$: $2.5-1.6 = 0.9$
• From $t=2$ to $t=3$: $3.9-2.5 = 1.4$
• From $t=3$ to $t=4$: $6.2-3.9 = 2.3$
• From $t=4$ to $t=5$: $9.8-6.2 = 3.6$

These differences are not constant; they are getting larger, so a simple increasing linear model (which needs constant differences) does not fit well.

For a quadratic model, the second differences (the differences of the differences) are roughly constant.

First differences (already found): $0.6, 0.9, 1.4, 2.3, 3.6$.

Second differences:

• $0.9 - 0.6 = 0.3$
• $1.4 - 0.9 = 0.5$
• $2.3 - 1.4 = 0.9$
• $3.6 - 2.3 = 1.3$

These second differences are not close to constant, so a quadratic model is not a good match either.

When values grow by adding about the same amount each step, that suggests one kind of model; when they grow by multiplying by about the same factor each step, that suggests a different model.

Compute the ratios of consecutive culture sizes:

• $1.6 / 1.0 = 1.6$
• $2.5 / 1.6 \approx 1.56$
• $3.9 / 2.5 = 1.56$
• $6.2 / 3.9 \approx 1.59$
• $9.8 / 6.2 \approx 1.58$

These ratios are all around $1.6$, meaning each hour the culture is multiplying by nearly the same factor, not just adding a fixed amount.

An increasing logarithmic model would have growth that slows down over time (the differences between terms would get smaller), but here the differences are getting larger and the growth speeds up. The data show values that increase faster and faster, with a nearly constant ratio between consecutive terms, which is characteristic of increasing exponential growth.

So the most appropriate model for $N$ as a function of $t$ is increasing exponential (choice C).