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

Answer 1

To model the exponential growth of the wasp population, we can use the exponential growth formula:

\[ N(t) = N_0 \cdot e^{rt} \]

where:

• \(N(t)\) is the population at time \(t\),
• \(N_0\) is the initial population,
• \(r\) is the growth rate,
• \(t\) is time (in months),
• \(e\) is the base of the natural logarithm (approximately equal to 2.71828).

In your case:

• The initial population \(N_0 = 5000\),
• The growth rate \(r = 0.13\) (which is 13% expressed as a decimal).

Substituting these values into the formula gives us:

\[ N(t) = 5000 \cdot e^{0.13t} \]

This equation models the growth of the wasp population over time, where \(t\) is in months.

Answer 2

As t increases, w increases slowly at first and then quickly
As t increases, w increases quickly at first then slowly
As t increases, w decreases slowly then quickly
As t increases, w decreases quickly then slowly

Answer 3

The behavior of the population growth of the wasps can be described as follows:

1. **As \( t \) increases, ** \( w \) (the wasp population) increases slowly at first and then quickly.

This characteristic is typical of exponential growth. Initially, when the population is small, the absolute increase in numbers is also small. However, as the population grows, the absolute number of wasps added within the same time period increases due to the larger population base, leading to faster growth.

So, the correct statement about the behavior of the wasp population over time is:

As \( t \) increases, \( w \) increases slowly at first and then quickly.