We are given the exponential function:

$y = 4000(0.51)^t$

We want to express this function in the form:

$y = ae^{kt}$

Step 1: Rewrite the function

We know the base $0.51$ can be converted into an exponential form. First, let's express $0.51$ as an exponential:

$0.51 = e^{\ln(0.51)}$

So, we can rewrite the function as:

$y = 4000 \cdot e^{\ln(0.51) \cdot t}$

Step 2: Identify the value of $k$

The exponential function is now:

$y = 4000 \cdot e^{(\ln(0.51)) \cdot t}$

Here, $k = \ln(0.51)$. Let's calculate $k$ to four decimal places.

$k = \ln(0.51) \approx -0.6733$

Thus, the function becomes:

$y = 4000 \cdot e^{-0.6733t}$

Step 3: Percent annual growth rate

The percent annual growth rate corresponds to the base $0.51$. Since $0.51$ represents a decay, we calculate it as:

$\text{Annual Growth Rate} = (0.51 - 1) \times 100 = -49\%$

This means the population decreases by 49% annually.

Step 4: Continuous percent growth rate

The continuous growth rate is determined by $k$. To express this as a percentage:

$\text{Continuous Growth Rate} = k \times 100 = -0.6733 \times 100 \approx -67.33\%$

Thus, the continuous decay rate is approximately $-67.33\%$ per year.

Summary

• The function in the form $y = ae^{kt}$ is: $y = 4000 \cdot e^{-0.6733t}$
• Percent annual growth rate: $-49\%$
• Continuous percent growth rate: $-67.33\%$

Would you like more details on any of these steps? Here are some related questions you might find interesting:

1. How does $k$ relate to the growth or decay rate in exponential functions?
2. What is the difference between the annual and continuous growth rates?
3. How can we model exponential decay in real-life scenarios?
4. What are some applications of continuous growth rates in economics?
5. How does the natural logarithm function help in converting between different exponential forms?

Tip: Exponential decay functions are often used in contexts like population decline, radioactive decay, or cooling processes!