A small nest of wasps has an exponential growth rate of 13% per month. If the nest currently has 5,000 wasps, the situation can be modeled by the equation w(t)=5,000(1.13)t

Question

, where w is the number of wasps after time t months. Which of the following statements is true about the equation?(1 point)
Responses

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

Answer 1

The equation provided models the growth of a population of wasps:

\[ w(t) = 5,000(1.13)^t \]

This is an exponential growth function, where the growth rate is 13% per month.

In exponential growth, as time \( t \) increases, the quantity \( w(t) \) increases at a rate proportional to its current size. This means:

1. Initially, when \( t \) is small (for example, \( t = 1 \) or \( t = 2 \)), the increase in \( w(t) \) will be relatively small because the base amount (5,000 wasps) is not very large.
2. However, as \( t \) increases, the value of \( w(t) \) becomes larger, and thus the increase (which is a percentage of the current value) also grows larger.

Therefore, the correct statement regarding the behavior of the function is:

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

This can be interpreted positively in the context of the growth. Thus, the correct choice from the responses provided is:

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