An Exponential Model with Unit Adjustment Show that the following data are exponential and find a formula for an exponential model. (Note: It will be necessary to make a unit adjustment. For this exercise, round your answers to three decimal places.)
t 0 4 8 
g(t) 38.30 28.65 21.43
t 12 16 20
g(t) 16.04 11.99 8.97

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.

An Exponential Model with Unit Adjustment: Show that the following data are exponential and find a formula for an exponential model. (Note: It will be necessary to make a unit adjustment. For this exercise, round your answers to three decimal places.) t 0 4 8, g(t) 38.30 28.65 21.43, t 12 16 20, g(t) 16.04 11.99 8.97.

Discover how to create an exponential decay model using the data points t = 0, 4, 8, 12, 16, 20 and g(t) = 38.30, 28.65, 21.43, 16.04, 11.99, 8.97. This step-by-step guide explains how to adjust for units and round to three decimal places. Suitable for Grades 10-12.

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