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

Question

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

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

Answer 1

To analyze the equation \( w(t) = 5,000(1.13)^t \) for the growth of the wasp nest, let's break down the statements provided regarding its domain and range:

1. Range (w-values):

   • Since the equation models wasp growth, \( w(t) \) represents the number of wasps, which cannot be negative. Therefore, the range starts from a positive number (greater than 0) and goes to infinity as \( t \) increases. Hence, the correct statement about the range is not "includes values greater than 0 and less than or equal to 5,000" but rather includes all values greater than 0.

2. Domain (t-values):

   • The variable \( t \) represents time in months, which cannot be negative. Therefore, \( t \) must be greater than or equal to 0 (\( t \geq 0 \)). So, the domain is correct to say that it includes values of \( t \) such that \( t \geq 0 \).

3. Other statements:

   • The range cannot be all real numbers since \( w(t) \) must be greater than 0, so the statement about the range being all real numbers is false.
   • The domain cannot be all real numbers since \( t \) is restricted to being 0 or positive.

From the analysis:

• The true statement is: The domain (t-values) includes the values of t such that t ≥ 0.