To define an appropriate domain for the situation described by the equation \( n(t) = 50 \times 1.2^{2t} \), where \( n \) is the number of mosquitoes and \( t \) is the time in days, let's consider the context and meaning of the variables:
-
Time (\( t \)): Since the growth of the mosquito population is being modeled from the starting point of zero days (i.e., the moment the observation begins), \( t \) should start at 0. Since we are interested in the growth over a period until the city sprays after 11 days, we need to limit \( t \) accordingly.
-
Population (\( n \)): The population \( n(t) \) represents the number of mosquitoes, which cannot be negative. However, since the domain for \( t \) is constrained by the events in the story (i.e., from starting point until the spraying), the population itself is inherently non-negative for all \( t \geq 0 \).
Inappropriate Domain:
Given these considerations, the inappropriate domain in this scenario would be:
- Negative Time: \( t < 0 \). Since time cannot be negative in this context, we cannot use negative values for \( t \).
- Time Greater than 11 Days: It may not be meaningful to model the growth of the mosquito population after the city has taken action to spray for mosquitoes, so \( t > 11 \) might also be considered inappropriate for modeling purposes.
Appropriate Domain:
Thus, the appropriate domain for the variable \( t \) that reflects the context of the problem is:
\[ t \in [0, 11] \]
This means \( t \) can take any value from 0 to 11, inclusive. Beyond this range, the model may not accurately depict the situation due to the intervention of the spraying.
1 answer