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

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)
Responses

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

GPT-4o mini · Answer

To analyze the given equation $ w(t) = 5,000(1.13)^t $ for the growth of wasps, let's evaluate the statements provided: 1. **Domain of $ t $**: - The variable $ t $ represents time in months and cannot be negative due to the context (you can’t have negative time). Therefore, the valid values for $ t $ are $ t \geq 0 $. - The correct statement here would be that the domain (t-values) includes the values of $ t $ such that $ t \geq 0 $. 2. **Range of $ w $**: - The function $ w(t) = 5,000(1.13)^t $ represents an exponential growth model, and as $ t $ increases, $ w(t) $ will continue to increase beyond 5,000. When $ t = 0 $, $ w(0) = 5,000 \cdot (1.13)^0 = 5,000 $. - The smallest value of $ w(t) $ occurs at $ t = 0 $ and is $ 5,000 $. As $ t $ increases, $ w(t) $ will approach infinity, so $ w $ will be greater than 5,000 indefinitely. - The range of $ w(t) $ is therefore $ w > 5,000 $, and it can never be less than or equal to 5,000. Based on this analysis, the true statements among the given options are: - The domain (t-values) includes the values of $ t $ such that $ t \geq 0 $. - The range (w-values) does not include all real numbers and is not limited to values less than or equal to 5,000; thus, the other options regarding the range are false. Therefore, the correct conclusion is: - **The domain (t-values) includes the values of t such that t ≥ 0**.