Let's analyze the given equation for the growth of the wasp population:
\[ w(t) = 5,000(1.13)^t \]
This equation models the population of wasps with respect to time \( t \), where:
- \( w(t) \) is the number of wasps after \( t \) months.
- The initial number of wasps is 5,000.
- The growth rate is 13% per month.
Domain Analysis:
The variable \( t \) represents time in months, and it cannot be negative because time cannot be less than zero in this context. Thus, the domain of \( t \) is:
- \( t \geq 0 \)
Range Analysis:
Next, let's consider the values of \( w(t) \):
- At \( t = 0 \): \( w(0) = 5,000(1.13)^0 = 5,000 \)
- As \( t \) increases, \( (1.13)^t \) grows without bound because \( 1.13 \) is greater than 1. Consequently, \( w(t) \) will increase indefinitely.
Thus, the range of \( w(t) \) includes values starting from 5,000 and going to infinity. Therefore, the range of \( w \) is:
- \( w > 5,000 \)
Evaluating the Statements:
-
The range (w-values) can be all real numbers: False
- The range is not all real numbers; it starts from 5,000 and goes to infinity.
-
The domain (t-values) includes the values of t such that t >= 0: True
- This is correct as explained above.
-
The domain (t-values) can be all real numbers: False
- The domain is restricted to \( t \geq 0 \).
-
The range (w-values) includes the values greater than 0 and less or equal to 5,000: False
- This is incorrect as the values of \( w(t) \) begin at 5,000 and grow without bound.
Conclusion:
The true statement about the equation is:
- The domain (t-values) includes the values of t such that t >= 0.