A warren of rabbits has a population of 25. The population is increasing at a rate of 20% per year. Select the model of the population after t years that reveals the monthly growth... A warren of rabbits has a population of 25. The population is increasing at a rate of 20% per year. Select the model of the population after t years that reveals the monthly growth rate.
Understand the Problem
The question is asking to determine which mathematical model accurately represents the growth of a rabbit population under a specified annual growth rate of 20%, converted to a monthly growth rate. This involves applying the formula for exponential growth and identifying the correct function among the given options.
Answer
$$ P(t) = P_0 e^{0.0153t} $$
Answer for screen readers
The mathematical model for the rabbit population growth can be expressed as
$$ P(t) = P_0 e^{0.0153t} $$
Steps to Solve
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Identify the annual growth rate
The problem states that the rabbit population grows at an annual rate of 20%, or 0.20 in decimal form. -
Convert the annual growth rate to monthly growth rate
To find the monthly growth rate from the annual growth rate, we use the formula:
$$ r = (1 + R)^{1/n} - 1 $$
where ( R ) is the annual growth rate and ( n ) is the number of compounding periods in a year.
Here, ( R = 0.20 ) and ( n = 12 ) (months in a year).
Calculating:
$$ r = (1 + 0.20)^{1/12} - 1 $$
$$ r = (1.20)^{1/12} - 1 $$
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Calculate the monthly growth rate
Now, we compute ( (1.20)^{1/12} ). Using a calculator or logarithmic function, we find:
$$ (1.20)^{1/12} \approx 1.0153 $$
Therefore,
$$ r \approx 1.0153 - 1 = 0.0153 $$
This implies that the monthly growth rate is approximately 1.53%. -
Write the exponential growth function
The formula for exponential growth can be represented as:
$$ P(t) = P_0 e^{rt} $$
Where:
- ( P(t) ) is the population at time ( t )
- ( P_0 ) is the initial population
- ( r ) is the monthly growth rate (0.0153)
- ( t ) is time in months
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Choose the correct option
Once the exponential function is established, we compare it against the given options to find the mathematical model that matches the form.
The mathematical model for the rabbit population growth can be expressed as
$$ P(t) = P_0 e^{0.0153t} $$
More Information
Exponential growth models are commonly used in ecological studies to represent populations. A monthly growth rate maintains accuracy in predictions over shorter periods, such as months, rather than years.
Tips
- Assuming a linear growth model instead of exponential when working with population growth.
- Not converting the annual rate correctly into a monthly rate or miscalculating the monthly rate.
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