The graph shows the deer population in a Pennsylvania county between 2003 and 2007. Assume that the population grows exponentially. (Assume 2003 is t = 0.) y intercept 10000 and point 4,26000. What is the growth rate? (Round your answer to the nearest whole number.

The problem indicates that the deer population grows exponentially, and the function modeling this population can be written as:

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

Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population (at \( t = 0 \)),
- \( r \) is the growth rate,
- \( t \) is time in years.

From the problem, we know:
- \( P_0 = 10,000 \) (initial population at \( t = 0 \)),
- \( P(4) = 26,000 \) (population after 4 years, at \( t = 4 \)).

We need to find the growth rate \( r \).

### Step 1: Set up the equation using the given values

From the exponential growth formula:

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

Substitute the values for \( P(4) = 26,000 \), \( P_0 = 10,000 \), and \( t = 4 \):

\[
26,000 = 10,000 e^{4r}
\]

### Step 2: Solve for \( r \)

First, divide both sides of the equation by 10,000 to isolate the exponential term:

\[
\frac{26,000}{10,000} = e^{4r}
\]
\[
2.6 = e^{4r}
\]

Now take the natural logarithm (ln) of both sides:

\[
\ln(2.6) = 4r
\]
\[
r = \frac{\ln(2.6)}{4}
\]

### Step 3: Calculate the value of \( r \)

Using a calculator:

\[
\ln(2.6) \approx 0.9555
\]
\[
r = \frac{0.9555}{4} \approx 0.2389
\]

Rounding to the nearest whole number percentage:

\[
r \times 100 \approx 24\%
\]

### Final Answer: The growth rate is approximately **24%** per year.

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the natural logarithm help in solving exponential equations?
2. What would happen to the growth rate if the population doubled in 4 years instead of increasing to 26,000?
3. How would you graph an exponential growth function for population modeling?
4. What if the population had been declining? How would that affect the equation?
5. Can you explain how exponential growth differs from linear growth?

**Tip:** Exponential growth means that as time passes, the rate of increase becomes faster, unlike linear growth, which remains constant over time.

The graph shows the deer population in a Pennsylvania county between 2003 and 2007. Assume that the population grows exponentially. (Assume 2003 is t = 0.) y intercept 10000 and point 4,26000. What is the growth rate? (Round your answer to the nearest whole number.)

Find the growth rate of a deer population in a Pennsylvania county between 2003 and 2007, modeled using exponential growth. Starting with a population of 10,000 in 2003 and growing to 26,000 by 2007, this problem involves using the exponential growth formula and logarithms. 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