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

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

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

Answer 1

To analyze the equation \( w(t) = 5,000(1.13)^t \) for the nest of wasps, let's break down the statements related to the range and domain of the function.

1. Range (w-values):

   • The function starts at \( 5,000 \) when \( t = 0 \) (i.e., \( w(0) = 5,000 \)) and increases exponentially due to the factor \( (1.13)^t \). As \( t \) increases, \( (1.13)^t \) grows larger, making \( w(t) \) increase indefinitely.
   • Therefore, the range of the function includes all values greater than or equal to \( 5,000 \). The minimum number of wasps (when \( t = 0 \)) is \( 5,000 \) and it approaches infinity as \( t \) increases.

2. Domain (t-values):

   • The time \( t \) cannot be negative in this context because you cannot have negative time in this real-world modeling situation. Therefore, \( t \) must be \( 0 \) or greater.
   • Thus, the domain includes values such that \( t \geq 0 \).

Combining these conclusions, the correct statement is:

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