a population grows from its initial level of A = 73000 at a continuous rate of r = 6.8% per year. 

a. Write a function, P(t), for the population in t years.

b. By what percent does the population grow each year? Round the answer to 3 decimal places. 

### Part (a): Writing the Population Function \( P(t) \)

Given:
- Initial population, \( A = 73,000 \)
- Continuous growth rate, \( r = 6.8\% \) per year

The formula for the population at any time \( t \) under continuous growth is given by:

\[
P(t) = A \cdot e^{rt}
\]

Substituting the given values:

\[
P(t) = 73{,}000 \cdot e^{0.068t}
\]

So, the function \( P(t) \) is:

\[
\boxed{P(t) = 73{,}000 \cdot e^{0.068t}}
\]

### Part (b): Calculating the Annual Growth Percentage

To find the equivalent annual growth rate, we use the relation between continuous growth rate and the equivalent discrete (annual) growth rate. If \( r_c \) is the continuous rate and \( r_a \) is the annual growth rate, they are related by:

\[
1 + r_a = e^{r_c}
\]

Substituting \( r_c = 0.068 \):

\[
1 + r_a = e^{0.068}
\]

Now, we calculate \( e^{0.068} \):

\[
1 + r_a = 1.070319
\]

Subtracting 1 to find \( r_a \):

\[
r_a = 1.070319 - 1 = 0.070319
\]

So, the population grows by:

\[
\boxed{7.032\%}
\]

each year, rounded to three decimal places.

---

Would you like further details or have any questions? 

### Related Questions
1. How would the function change if the growth rate were compounded annually instead of continuously?
2. What will the population be in 5 years?
3. How long will it take for the population to double?
4. What is the effect of doubling the initial population on the growth function?
5. How does the growth rate affect the time it takes to reach a population of 150,000?
6. Can you derive the formula for continuous growth from the compound interest formula?
7. What would be the population after 10 years if the growth rate changes to 8%?
8. How does the population function change if the growth rate is negative?

### Tip
Always round your final answers according to the problem’s requirements to ensure precision in communication.

Explore continuous population growth with an initial population of 73,000 growing at a rate of 6.8% per year. Understand the population function P(t) = 73,000 * e^(0.068t) and calculate the annual growth rate of 7.032%. Learn how continuous compounding affects population growth over time.

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