To determine if the given data can be modeled by an exponential function, we will first attempt to find a formula for the exponential model $g(t) = g_0 e^{kt}$, where $g_0$ is the initial value (the value of $g(t)$ when $t = 0$), and $k$ is the growth/decay rate.

Step 1: Checking for exponential decay

Given the data points:

$t$	$g(t)$
0	38.30
4	28.65
8	21.43
12	16.04
16	11.99
20	8.97

We expect the data to follow the model $g(t) = g_0 e^{kt}$, where $g_0 = 38.30$.

Step 2: Calculating the decay rate $k$

We will use two points to solve for $k$. Let's choose the points $t = 0$ and $t = 4$ to set up the equation:

$g(4) = g_0 e^{4k}$

Substitute the known values $g(4) = 28.65$ and $g_0 = 38.30$:

$28.65 = 38.30 e^{4k}$

Solve for $k$:

$\frac{28.65}{38.30} = e^{4k}$ $0.7480 = e^{4k}$ $4k = \ln(0.7480)$ $4k \approx -0.2894$ $k \approx -0.07235$

Step 3: Writing the exponential model

Now that we have $k \approx -0.07235$ and $g_0 = 38.30$, we can write the exponential model as:

$g(t) = 38.30 e^{-0.07235t}$

Step 4: Unit adjustment and model verification

To verify if this model is appropriate, we will check it against other points in the data. For example, when $t = 8$, using our model:

$g(8) = 38.30 e^{-0.07235 \times 8} = 38.30 e^{-0.5788} \approx 38.30 \times 0.5610 \approx 21.47$

This is very close to the actual value of $g(8) = 21.43$, confirming that the model is accurate.

Final Exponential Model:

$g(t) = 38.30 e^{-0.072t}$

This formula represents the exponential decay for the given data.

---

Would you like more details on the steps or explanations on the unit adjustment? Here are some related questions:

1. How can we calculate the decay constant $k$ using different points from the data set?
2. What role does the natural logarithm play in solving exponential equations?
3. How can we use this model to predict values for $g(t)$ at times not included in the data set?
4. What happens to the function as $t \to \infty$ in this exponential decay model?
5. How would we adjust the model if the data represented exponential growth instead of decay?

Tip: Always verify the accuracy of an exponential model by testing it with multiple points from your data set.