The population of a colony of rabbits grows exponentially. The colony begins with 20 rabbits; 5 years later there are 390 rabbits.
(a) Express the population of the colony of rabbits, 𝑃, as a function of time, 𝑡, in years.
𝑃(𝑡)=

The growth of the rabbit population follows an exponential model, which can be expressed as:

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

Where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population,
- \(k\) is the growth rate constant, and
- \(t\) is the time in years.

### Step 1: Use given values to find \(P_0\) and set up the equation.
- The initial population \(P_0\) is 20 rabbits, so:
  \[
  P(t) = 20 e^{kt}
  \]

### Step 2: Use the second condition to find \(k\).
- After 5 years, the population is 390 rabbits:
  \[
  P(5) = 390
  \]
  Substitute this into the equation:
  \[
  390 = 20 e^{5k}
  \]

### Step 3: Solve for \(k\).
First, divide both sides by 20:
\[
e^{5k} = \frac{390}{20} = 19.5
\]

Now, take the natural logarithm of both sides:
\[
5k = \ln(19.5)
\]
\[
k = \frac{\ln(19.5)}{5}
\]

Now calculate \(k\):
\[
k \approx \frac{2.9704}{5} \approx 0.5941
\]

### Step 4: Write the function \(P(t)\).
Now that we have \(k \approx 0.5941\), the population function is:
\[
P(t) = 20 e^{0.5941t}
\]

This is the exponential model for the population of the colony of rabbits as a function of time.

Would you like to see the details of solving for \(k\), or do you have any further questions?

---

Here are 5 related questions:
1. How would the equation change if the initial population was 50 instead of 20?
2. How long would it take for the population to double using this model?
3. Can you solve for the population after 10 years?
4. What is the population growth rate as a percentage?
5. How would the model change if the growth rate decreased by half?

**Tip:** Always check the units of time when applying exponential growth formulas to ensure consistency throughout your calculations.

The population of a colony of rabbits grows exponentially. The colony begins with 20 rabbits; 5 years later there are 390 rabbits. (a) Express the population of the colony of rabbits, P, as a function of time, t, in years.

Learn how to model the exponential growth of a rabbit population, starting with 20 rabbits and growing to 390 in 5 years. This problem involves key mathematical concepts such as exponential growth and logarithms, and uses the formula P(t) = 20e^(0.5941t) to express the population as a function of time. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation