Question
Anna wants to make banana bread. She buys bananas and noticed fruit flies after she got home. The number of fruit flies grows at an exponential rate modeled by f(t) = 8(2)^t
Where f is the number of fruit flies after t days Define the appropriate domain for the problem if Anna bakes the banana bread 5 days after shopping
0<=t<=5
0<=f<=256
0<=f<=5
0<=t<=256
Where f is the number of fruit flies after t days Define the appropriate domain for the problem if Anna bakes the banana bread 5 days after shopping
0<=t<=5
0<=f<=256
0<=f<=5
0<=t<=256
Answers
Answered by
GPT-4o mini
Given the scenario, Anna wants to model the number of fruit flies over time as she prepares to bake banana bread. The function \( f(t) = 8(2)^t \) describes the number of fruit flies after \( t \) days, with the condition that she bakes the banana bread 5 days after shopping.
1. **Time Domain**: Since Anna bakes the banana bread 5 days after shopping, the time \( t \) should be within the interval from 0 to 5 days. This provides us with the first part of the domain, which is:
\[
0 \leq t \leq 5
\]
2. **Value Range for \( f \)**: To find the range of values for \( f(t) \) when \( t \) is from 0 to 5, we will compute \( f(0) \) and \( f(5) \):
- At \( t = 0 \):
\[
f(0) = 8(2)^0 = 8 \cdot 1 = 8
\]
- At \( t = 5 \):
\[
f(5) = 8(2)^5 = 8 \cdot 32 = 256
\]
Therefore, as \( t \) ranges from 0 to 5, \( f(t) \) will range from 8 to 256. Thus, we can summarize the range for \( f \):
\[
8 \leq f \leq 256
\]
3. **Other Options**: Analyzing the provided options:
- \( 0 \leq f \leq 256 \): This is partially correct because \( f \) starts at 8, but the range from zero to 256 does not reflect the actual values \( f(t) \) will take.
- \( 0 \leq f \leq 5 \): This is incorrect because \( f(t) \) starts at 8 and goes up to 256.
- \( 0 \leq t \leq 256 \): This is also incorrect since \( t \) can only go up to 5.
Based on this analysis, **the appropriate domain for the problem is**:
\[
0 \leq t \leq 5
\]
And the range of \( f \) is:
\[
8 \leq f \leq 256
\]
1. **Time Domain**: Since Anna bakes the banana bread 5 days after shopping, the time \( t \) should be within the interval from 0 to 5 days. This provides us with the first part of the domain, which is:
\[
0 \leq t \leq 5
\]
2. **Value Range for \( f \)**: To find the range of values for \( f(t) \) when \( t \) is from 0 to 5, we will compute \( f(0) \) and \( f(5) \):
- At \( t = 0 \):
\[
f(0) = 8(2)^0 = 8 \cdot 1 = 8
\]
- At \( t = 5 \):
\[
f(5) = 8(2)^5 = 8 \cdot 32 = 256
\]
Therefore, as \( t \) ranges from 0 to 5, \( f(t) \) will range from 8 to 256. Thus, we can summarize the range for \( f \):
\[
8 \leq f \leq 256
\]
3. **Other Options**: Analyzing the provided options:
- \( 0 \leq f \leq 256 \): This is partially correct because \( f \) starts at 8, but the range from zero to 256 does not reflect the actual values \( f(t) \) will take.
- \( 0 \leq f \leq 5 \): This is incorrect because \( f(t) \) starts at 8 and goes up to 256.
- \( 0 \leq t \leq 256 \): This is also incorrect since \( t \) can only go up to 5.
Based on this analysis, **the appropriate domain for the problem is**:
\[
0 \leq t \leq 5
\]
And the range of \( f \) is:
\[
8 \leq f \leq 256
\]