As time goes on, what happens to $e^{-0.6t}$ when $t$ becomes a large positive number? How does that affect the denominator and the overall value of $S(t)$?

Imagine the graph of $y=S(t)$. As $t$ increases, does the graph keep rising without bound, level off, or decrease? What special horizontal line does it get closer to, and what does that line represent in the context?

Check whether $75$ can be the initial value, the value at $t=30$, or the total increase by doing quick calculations for $S(0)$ and $S(30)$ and thinking about the change over the interval.

Type S(t) = 75/(1+4*e^(-0.6*t)) into Desmos. If needed, change the variable to x (for example, S(x) = 75/(1+4*e^(-0.6*x))).

Use the table feature or tap on the graph at $t=0$ and $t=30$ to see the corresponding $y$-values. Compare those values to $75$ to decide whether $75$ is the initial value, the value at 30 minutes, or something else.

Zoom out horizontally to the right (increasing $t$). Notice the horizontal line that the graph gets closer to but does not cross. Read off the $y$-value of this horizontal level and relate that behavior to the answer choices.

The function is

As $t$ increases, the term $e^{-0.6t}$ gets smaller because the exponent is negative. This means the denominator $1 + 4e^{-0.6t}$ decreases toward $1$, and the whole fraction increases toward some upper bound. The constant $75$ is in the numerator and will be related to that upper bound.

Evaluate the function at $t=0$:

So immediately after the beverage is consumed, the model predicts a sugar concentration of $15$ grams per liter, not $75$. This rules out any interpretation that says $75$ is the initial or minimum value.

Now think about what happens as $t$ gets very large:

• The exponent $-0.6t$ becomes a large negative number.
• So $e^{-0.6t}$ becomes very close to $0$.

That makes the denominator approach $1 + 4\cdot 0 = 1$, so

This means the graph of $y=S(t)$ gets closer and closer to the horizontal line $y=75$ but does not go above it. That line is the horizontal asymptote and represents a limiting (maximum) value the model approaches over time.

From the work above:

• $75$ is not the value at $t=0$ (that is $15$), so it is not the minimum.
• $75$ is not exactly the value at $t=30$; it is just a little less than $75$ because $e^{-0.6\cdot 30}$ is very small but not zero.
• $75$ is also not the total increase; the increase from $t=0$ to large $t$ is about $75 - 15 = 60$.

Instead, $75$ is the horizontal asymptote, the upper limit that $S(t)$ gets closer to as time passes. Therefore, the correct interpretation is the maximum sugar concentration the model predicts the bloodstream will approach as time passes.