Question
f(t)=7(2)^t where f is the number of flies and t is the number of days define an appropriate domain for the problem after 5 days
0<=t<=5
O<=f<=256
0<=f<=5
0<=t<=256
0<=t<=5
O<=f<=256
0<=f<=5
0<=t<=256
Answers
GPT-4o mini
Given the function \( f(t) = 7(2)^t \), where \( f \) represents the number of flies and \( t \) represents the number of days, we need to define an appropriate domain for the problem after 5 days.
1. **Time domain \( t \)**: Since we're looking at the number of flies over the course of days, and we want to consider it after 5 days, our domain for \( t \) should start at 0 (initially at day 0) and go up to 5. Thus, the appropriate domain for \( t \) is:
\[
0 \leq t \leq 5
\]
2. **Fly count \( f \)**: To find the range of \( f \) after 5 days, we can evaluate \( f(t) \) at the endpoints of the domain:
- At \( t = 0 \):
\[
f(0) = 7(2)^0 = 7
\]
- At \( t = 5 \):
\[
f(5) = 7(2)^5 = 7 \times 32 = 224
\]
Therefore, as \( t \) varies from 0 to 5, \( f(t) \) will vary from 7 to 224. Thus, the appropriate range for \( f \) is:
\[
7 \leq f \leq 224
\]
### Summary
- Domain of \( t \): \( 0 \leq t \leq 5 \)
- Range of \( f \): \( 7 \leq f \leq 224 \)
None of the options provided ( \( O \leq f \leq 256 \), \( 0 \leq f \leq 5 \), \( 0 \leq t \leq 256 \) ) are entirely appropriate based on the analysis of the function. The correct interpretation should keep the \( t \) restricted between 0 and 5 and \( f \) between 7 and 224 based on the initially defined function.
1. **Time domain \( t \)**: Since we're looking at the number of flies over the course of days, and we want to consider it after 5 days, our domain for \( t \) should start at 0 (initially at day 0) and go up to 5. Thus, the appropriate domain for \( t \) is:
\[
0 \leq t \leq 5
\]
2. **Fly count \( f \)**: To find the range of \( f \) after 5 days, we can evaluate \( f(t) \) at the endpoints of the domain:
- At \( t = 0 \):
\[
f(0) = 7(2)^0 = 7
\]
- At \( t = 5 \):
\[
f(5) = 7(2)^5 = 7 \times 32 = 224
\]
Therefore, as \( t \) varies from 0 to 5, \( f(t) \) will vary from 7 to 224. Thus, the appropriate range for \( f \) is:
\[
7 \leq f \leq 224
\]
### Summary
- Domain of \( t \): \( 0 \leq t \leq 5 \)
- Range of \( f \): \( 7 \leq f \leq 224 \)
None of the options provided ( \( O \leq f \leq 256 \), \( 0 \leq f \leq 5 \), \( 0 \leq t \leq 256 \) ) are entirely appropriate based on the analysis of the function. The correct interpretation should keep the \( t \) restricted between 0 and 5 and \( f \) between 7 and 224 based on the initially defined function.