A town starts with 30,000 residents and grows at a continuous rate of 7% per year.

(a) Write a formula for the population of the town after t years.

(b) What is the population of the town after 8 years have passed?

(c) Find the annual growth rate of the population of the town.

The town's population growth can be modeled using the formula for continuous exponential growth, which is:

\[
P(t) = P_0 \cdot e^{rt}
\]

where:
- \( P(t) \) is the population after \( t \) years,
- \( P_0 \) is the initial population,
- \( r \) is the continuous growth rate (as a decimal),
- \( t \) is the time in years,
- \( e \) is Euler's number (approximately 2.71828).

### (a) Formula for the population of the town after \( t \) years:
Given:
- Initial population, \( P_0 = 30,000 \),
- Continuous growth rate, \( r = 0.07 \) (since 7% = 0.07 in decimal form),

The formula for the population after \( t \) years is:

\[
P(t) = 30,000 \cdot e^{0.07t}
\]

### (b) Population after 8 years:
We can use the formula \( P(t) = 30,000 \cdot e^{0.07t} \) to calculate the population after 8 years, where \( t = 8 \).

\[
P(8) = 30,000 \cdot e^{0.07 \cdot 8}
\]

Let's calculate this:

\[
P(8) = 30,000 \cdot e^{0.56}
\]

We can now find the exact value of \( P(8) \).

### (c) Annual growth rate of the population:
The continuous growth rate given is 7% per year. To find the **annual growth rate**, we can use the relationship between continuous and annual growth rates:

\[
r_{\text{annual}} = e^{r_{\text{continuous}}} - 1
\]

Substituting \( r_{\text{continuous}} = 0.07 \):

\[
r_{\text{annual}} = e^{0.07} - 1
\]

Now, I'll compute the exact values for both parts (b) and (c).### (b) Population after 8 years:
The population of the town after 8 years is approximately:

\[
P(8) \approx 52,520 \text{ residents}
\]

### (c) Annual growth rate:
The annual growth rate is approximately:

\[
7.25\% \text{ per year}
\]

Do you want further details or have any questions? Here are five related questions to explore:

1. How would the population change if the growth rate were 5% instead of 7%?
2. How long will it take for the population to double at this growth rate?
3. What is the population after 15 years?
4. How does continuous growth compare to simple interest growth over 10 years?
5. How do exponential growth models apply to population dynamics in real-world scenarios?

**Tip**: For long-term growth predictions, exponential models provide more accuracy, especially for natural growth processes.

A town starts with 30,000 residents and grows at a continuous rate of 7% per year. (a) Write a formula for the population of the town after t years. (b) What is the population of the town after 8 years have passed? (c) Find the annual growth rate of the population of the town.

Explore how to model the population growth of a town starting with 30,000 residents growing at a continuous rate of 7% per year. This example covers exponential growth formulas and population projections, including calculating the town's population after 8 years and determining the annual growth rate. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation