Question
A small nest of wasps has an exponential growth rate of 13% per month. If the nest currently has 5000 wasps, the situation can be modeled by which equation?
Answers
GPT-4o mini
To model the growth of the wasp nest, we can use the formula for exponential growth:
\[
N(t) = N_0 \cdot e^{rt}
\]
where:
- \( N(t) \) is the number of wasps at time \( t \),
- \( N_0 \) is the initial number of wasps,
- \( r \) is the growth rate,
- \( t \) is the time (in months),
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
In your case:
- The initial number of wasps, \( N_0 \), is 5000.
- The monthly growth rate is 13%, which can be expressed as a decimal: \( r = 0.13 \).
Substituting these values into the exponential growth formula gives:
\[
N(t) = 5000 \cdot e^{0.13t}
\]
This equation models the growth of the wasp nest over time \( t \) measured in months.
\[
N(t) = N_0 \cdot e^{rt}
\]
where:
- \( N(t) \) is the number of wasps at time \( t \),
- \( N_0 \) is the initial number of wasps,
- \( r \) is the growth rate,
- \( t \) is the time (in months),
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
In your case:
- The initial number of wasps, \( N_0 \), is 5000.
- The monthly growth rate is 13%, which can be expressed as a decimal: \( r = 0.13 \).
Substituting these values into the exponential growth formula gives:
\[
N(t) = 5000 \cdot e^{0.13t}
\]
This equation models the growth of the wasp nest over time \( t \) measured in months.