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

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

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) includes the values of t such that t≥0
.

Answer 1

To analyze the equation \( w(t) = 5,000(1.13)^t \), let's consider the domain and range.

1. Domain:

   • The variable \( t \) represents time in months. Since time cannot be negative, the domain of \( t \) is non-negative.
   • Therefore, the correct statement is: The domain (t-values) includes the values of t such that \( t \geq 0 \).

2. Range:

   • The equation models the number of wasps, \( w(t) \).
   • Since \( (1.13)^t \) is always positive, and the base value is 5,000, \( w(t) \) will always be greater than 0.
   • As \( t \) approaches infinity, \( w(t) \) increases without upper bound. Thus, the range is all values greater than 0.
   • Therefore, the statement, "The range (w-values) includes the values greater than 0 and less than or equal to 5,000" is incorrect. The range is actually all values greater than 0.

Considering this information, the correct statement is: The domain (t-values) includes the values of t such that t ≥ 0.