Compute the change in $C$ between each pair of consecutive times. Are those changes about the same, or do they keep increasing or decreasing?

After looking at differences, also compare the ratio of each $C$ value to the one before it. Which seems more consistent: the differences or the ratios?

In Desmos, add a table. Enter the time values $t$ in the $x_1$ column (0, 1, 2, 3, 4) and the concentration values $C$ in the $y_1$ column (5.0, 7.3, 10.7, 15.7, 23.1).

In a new expression line, type y1 ~ m x1 + b to perform a linear regression. Look at how well the line fits the plotted points and note the $R^2$ value (closer to 1 means a better fit).

In another line, type y1 ~ a x1^2 + b x1 + c to perform a quadratic regression. Compare this curve’s fit and $R^2$ value with the linear model.

Next, type y1 ~ a b^x1 to perform an exponential regression. Compare how closely this curve passes through the points and its $R^2$ value to the previous models; identify which model type has the best fit.

Look at the $t$ values in the table: $0,1,2,3,4$. They increase by $1$ hour each time, so the time intervals are equal. This lets us compare how the concentration $C$ changes over equal time steps.

Find how much $C$ increases each hour.

• From $t=0$ to $t=1$: $7.3-5.0=2.3$
• From $t=1$ to $t=2$: $10.7-7.3=3.4$
• From $t=2$ to $t=3$: $15.7-10.7=5.0$
• From $t=3$ to $t=4$: $23.1-15.7=7.4$

These increases are not constant; they are getting larger. That means a linear model (which needs a constant difference) is not appropriate. Also, the second differences (like $3.4-2.3$, $5.0-3.4$, $7.4-5.0$) are not constant either, so it does not fit a neat quadratic pattern.

Now compare ratios of consecutive $C$ values.

• $\dfrac{7.3}{5.0} \approx 1.46$
• $\dfrac{10.7}{7.3} \approx 1.47$
• $\dfrac{15.7}{10.7} \approx 1.47$
• $\dfrac{23.1}{15.7} \approx 1.47$

These ratios are all very close to one another, meaning each hour the concentration is multiplied by almost the same factor (about $1.47$). This is a strong sign of a multiplicative growth pattern instead of an additive one.

On the SAT, remember these quick checks:

• Linear: equal $t$ steps give equal differences in $C$.
• Quadratic: equal $t$ steps give differences in $C$ that change at a constant rate (second differences constant).
• Logarithmic: increases are large at first and then slow down over time.
• Exponential: equal $t$ steps give $C$ values that change by a nearly constant ratio (multiply by about the same factor each time).

Here, the differences are not constant, the second differences are not constant, and the increases are getting larger, not smaller. But the ratios between consecutive $C$ values are nearly constant, so the most appropriate model is exponential (choice B).