Find an exponential function of the form

Upper P left parenthesis t right parenthesis equals Upper P 0 n Superscript StartFraction t Over Upper T EndFraction

that models the​ situation, and then find the equivalent exponential model of the form

Upper P left parenthesis t right parenthesis equals Upper P 0 e Superscript rt

.

Doubling time of

4

​yr, initial population of

500

.

Let's solve this step by step.

### 1. Exponential function in the form \( P(t) = P_0 n^{\frac{t}{T}} \)

We are given:
- **Doubling time** \( T = 4 \) years,
- **Initial population** \( P_0 = 500 \).

The general form is:
\[
P(t) = P_0 \cdot n^{\frac{t}{T}}
\]
Since the population doubles every 4 years, \( n = 2 \). So we plug in the values:

\[
P(t) = 500 \cdot 2^{\frac{t}{4}}
\]

This is the exponential model of the form \( P(t) = P_0 n^{\frac{t}{T}} \).

### 2. Equivalent exponential model of the form \( P(t) = P_0 e^{rt} \)

To convert the function into the form \( P(t) = P_0 e^{rt} \), we need to express the base 2 exponential as an equivalent base \( e \) exponential.

Start from the function we found:
\[
P(t) = 500 \cdot 2^{\frac{t}{4}}
\]

To convert this to the base \( e \), we use the identity:
\[
a^x = e^{x \ln a}
\]
In our case, \( a = 2 \) and \( x = \frac{t}{4} \), so:

\[
2^{\frac{t}{4}} = e^{\frac{t}{4} \ln 2}
\]

Thus, the function becomes:
\[
P(t) = 500 \cdot e^{\frac{t}{4} \ln 2}
\]

Now we compare this with the desired form \( P(t) = P_0 e^{rt} \), and we can see that:
\[
r = \frac{\ln 2}{4}
\]

### Final answers:

- The exponential model in the form \( P(t) = P_0 n^{\frac{t}{T}} \) is:
  \[
  P(t) = 500 \cdot 2^{\frac{t}{4}}
  \]
  
- The equivalent exponential model in the form \( P(t) = P_0 e^{rt} \) is:
  \[
  P(t) = 500 \cdot e^{\frac{\ln 2}{4} t}
  \]

Would you like any further clarification or details?

---

### Related questions:
1. How does the doubling time affect the exponential growth rate \( r \)?
2. What is the general process for converting between exponential forms with different bases?
3. Can this model be used to calculate population at specific times, such as after 10 years?
4. How does changing the doubling time \( T \) alter the growth rate in the model?
5. What is the relationship between the base \( n \) and the growth rate \( r \) in these models?

### Tip: 
For growth models, remember that the base \( e \) is often preferred in continuous models because it simplifies differentiation and integration tasks.

Find an exponential function of the form P(t) = P0 * n^(t/T) that models the situation, and then find the equivalent exponential model of the form P(t) = P0 * e^(rt). Doubling time of 4 years, initial population of 500.

Learn how to model population growth with exponential functions using a doubling time of 4 years and an initial population of 500. This tutorial covers key exponential formulas and conversion to the natural base e. Suitable for Grades 10-12 students.

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