Question

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

Answer 1

To analyze the behavior of the equation \( w(t) = 5,000(1.13)^t \), where \( w \) is the number of wasps after \( t \) months, we note the following:

1. The term \( (1.13)^t \) indicates that the population is growing exponentially, not decreasing. As time \( t \) increases, the value of \( (1.13)^t \) also increases.

2. An exponential growth function initially increases slowly, but as \( t \) gets larger, the rate of increase becomes quicker due to the exponential nature of the function.

Therefore, the correct statement regarding the equation is:

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