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 t months. Which of the following statements is true about the equation?(1 point)
Responses

The domain (t-values) includes the values of t such that t≥0
.
The domain ( t -values) includes the values of t such that t greater-than-or-equal-to 0 .

The range (w-values) includes the values greater than 0 and less than or equal to 5,000.
The range ( w -values) includes the values greater than 0 and less than or equal to 5,000.

The domain (t-values) can be all real numbers.
The domain ( t -values) can be all real numbers.

The range (w-values) can be all real numbers.

Answer 1

To analyze the given model \( w(t) = 5,000(1.13)^t \), we need to determine the domain and range of the function.

1. Domain (t-values):

   • The variable \( t \) represents time in months and cannot be negative, meaning that \( t \) can only take on values of 0 and greater.
   • Therefore, the correct statement about the domain is:
     • "The domain (t-values) includes the values of t such that t ≥ 0."

2. Range (w-values):

   • The function \( w(t) = 5,000(1.13)^t \) starts at 5,000 when \( t = 0 \) and increases exponentially as \( t \) increases. This means that \( w(t) \) will always be greater than 5,000 (as 1.13 raised to any positive power will yield a number greater than 1).
   • Thus, the range of \( w \) is \( w > 0 \) and can be infinitely large, but it will never drop below 5,000.
   • The correct statement about the range is:
     • "The range (w-values) includes values greater than 0."

3. As for the remaining options:

   • The statement saying that the domain can be all real numbers is incorrect because \( t \) cannot be negative.
   • Similarly, the assertion that the range can be all real numbers is also incorrect, as the number of wasps cannot be negative and will always be more than 5,000 as time progresses.

In summary, the true statements from the options provided are:

• "The domain (t-values) includes the values of t such that t ≥ 0."