https://file.aiquickdraw.com/mathbot/file/17294817351805a3pusn2.jpg

We are given the population model:

\[
P(t) = 400(1.065)^t
\]

where \( P(t) \) is the population in thousands, and \( t \) is the number of years since 2000. We need to find how fast the population is changing in the year 2013, which corresponds to \( t = 2013 - 2000 = 13 \).

To find the rate at which the population is changing, we need to calculate the derivative of \( P(t) \) with respect to \( t \), which represents the rate of change of the population. The formula for the derivative of an exponential function is:

\[
P'(t) = \frac{d}{dt} \left( 400(1.065)^t \right)
\]

Using the chain rule, the derivative is:

\[
P'(t) = 400 \cdot \ln(1.065) \cdot (1.065)^t
\]

Now, substitute \( t = 13 \) to find the rate of change in 2013:

\[
P'(13) = 400 \cdot \ln(1.065) \cdot (1.065)^{13}
\]

First, calculate \( \ln(1.065) \approx 0.0631 \), and \( (1.065)^{13} \approx 2.2815 \).

\[
P'(13) = 400 \cdot 0.0631 \cdot 2.2815 \approx 400 \cdot 0.1439 \approx 57.57
\]

Since the population is in thousands, the rate of change is approximately:

\[
P'(13) \approx 57.57 \times 1000 \approx 57,570 \text{ people per year}.
\]

So, the closest answer to this calculation is **57,118 people per year**.

Do you want any more details or have any questions?

Here are 5 related questions:
1. How do we compute the derivative of other types of population models?
2. What does the rate of change of a population tell us about its growth?
3. How would a change in the base of the exponential function (e.g., 1.065) affect the population growth rate?
4. What would happen to the population if the growth rate were to decrease over time?
5. Can exponential growth continue indefinitely, or are there practical limitations?

**Tip:** In exponential growth models, the base of the exponential function controls the growth rate. A base greater than 1 indicates growth, while a base less than 1 indicates decay.

Suppose a population can be modeled by P(t) = 400(1.065)^t, where P(t) is in thousands and t is years since 2000. How fast is the population changing in 2013?

Learn how to calculate the rate of change for an exponential population growth model P(t) = 400(1.065)^t at t = 13 (year 2013). This problem involves finding the derivative of an exponential function using the chain rule and analyzing population growth over time. Suitable for high school students in 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