Question

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 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.
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Answer 1

The equation given is \( w(t) = 5,000(1.13)^t \). This represents exponential growth, where the growth rate is 13% per month.

In exponential growth situations, the quantity increases rapidly as time passes. In the context of your question, this means:

• As t increases, w increases quickly at first and then slows down: This statement is inaccurate for exponential growth.
• As t increases, w decreases quickly at first and then slowly: This statement is about decay, not growth.
• As t increases, w decreases slowly at first and then quickly: Again, this statement describes decay.
• As t increases, w increases slowly at first and then quickly: This is the correct choice; in exponential growth, the quantity does start increasing slowly but then accelerates significantly over time.

Therefore, the true statement is:

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

Answer 2

the t on "w(t)=5,000(1.13)t" was actually like this w(t)=5,000(1.13)^t

Answer 3

Thank you for clarifying that the correct equation is \( w(t) = 5,000(1.13)^t \).

This change doesn't affect the analysis much, as the equation still represents exponential growth with a base of \( 1.13 \).

In this case, the growth can be interpreted as follows:

• When \( t = 0 \), \( w(0) = 5,000(1.13)^0 = 5,000 \).
• As \( t \) increases, \( (1.13)^t \) increases exponentially.

The correct observation in exponential growth is that:

• As t increases, \( w \) increases slowly at first and then quickly. This is typical for exponential functions: they grow slowly initially and then the growth accelerates, leading to very rapid increases.

Thus, the true statement is:

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