Given the exponential growth model:

$V(t) = V_0 e^{kt}$

where:

• $V(t)$ is the value of the coin at time $t$.
• $V_0 = 282$ is the initial value of the coin in 1980.
• $k$ is the exponential growth rate.
• $t$ is the time in years since 1980.

The coin was sold again in 1986, so $t = 1986 - 1980 = 6$. The value at that time was $V(6) = 430$. We need to find the growth rate $k$.

Plugging the known values into the formula:

$430 = 282 e^{6k}$

To solve for $k$, follow these steps:

1. Divide both sides by 282:

   $\frac{430}{282} = e^{6k}$

2. Simplify the fraction:

   $1.5248 \approx e^{6k}$

3. Take the natural logarithm of both sides:

   $\ln(1.5248) = 6k$

4. Solve for $k$:

   $k = \frac{\ln(1.5248)}{6}$

Let’s calculate $k$ to the nearest thousandth.The value of the exponential growth rate $k$ is approximately $0.07$ (rounded to the nearest thousandth).

Would you like further details or have any questions?

Here are some related questions:

1. How would the value change if the coin was sold in 1990 instead?
2. What would the value of the coin be in 2024 if the same growth rate continued?
3. How can you determine the doubling time for the coin’s value?
4. What does an exponential growth rate of $k = 0.07$ imply in terms of percentage growth per year?
5. How would the model change if the growth was linear instead of exponential?

Tip: Exponential growth models are useful for predicting future values when growth rates are consistent over time.