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?Which of the following statements is true about the above wasp equation?

(1 point)
Responses

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

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

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

As t increases, w decreases quickly at first and then slowly
As t increases, w decreases quickly at first and then slowly
Question 3

1 answer

To model the growth of the wasp nest with an exponential growth rate, we can use the formula:

\[ w(t) = w_0 \cdot e^{rt} \]

Where:

  • \( w(t) \) is the population at time \( t \),
  • \( w_0 \) is the initial population (5000 wasps),
  • \( r \) is the growth rate (13% per month, or 0.13 as a decimal),
  • \( t \) is the time in months,
  • \( e \) is the base of the natural logarithm.

Substituting the values into the equation, we get:

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

Analyzing the Growth:

For the statements about the growth behavior:

  1. As t increases, w increases slowly at first and then quickly - This statement is true for exponential growth. In the beginning, the population increases slowly because the base (initial population) is small relative to its growth potential. However, as \( t \) increases and the population grows, the effect of the growth rate leads to increasingly larger increases.

  2. As t increases, w increases quickly at first and then slowly - This statement does not accurately describe exponential growth.

  3. As t increases, w decreases slowly at first and then quickly - This statement is not applicable because the population is growing, not decreasing.

  4. As t increases, w decreases quickly at first and then slowly - This statement is also not applicable for the same reason as above.

Conclusion:

The correct statement about the wasp population is:

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