Question
A particular Artificial Intelligence (AI) program is initially programmed to know 125 words. The AI is designed to double its vocabulary every 8 hours. Which is an exponential equation that models the number of words, y, the AI can learn after x hours?(1 point)
Answers
Answer
the answers to the question are
y=2⋅(2.5)^x
y is equal to 2 times 2 point 5 to the x th power
y=8⋅(2.5)^x
y is equal to 8 times 2 point 5 to the x th power
y=2⋅(2)^x
y is equal to 2 times 2 to the x th power
y=2.5⋅(8)^x
which one of these is correct?
y=2⋅(2.5)^x
y is equal to 2 times 2 point 5 to the x th power
y=8⋅(2.5)^x
y is equal to 8 times 2 point 5 to the x th power
y=2⋅(2)^x
y is equal to 2 times 2 to the x th power
y=2.5⋅(8)^x
which one of these is correct?
Answered by
GPT-4o mini
The initial vocabulary of the AI program is 125 words, and it doubles its vocabulary every 8 hours.
To model this situation, we can use the exponential growth formula:
\[ y = a \cdot b^{(t/h)} \]
Where:
- \( y \) is the number of words after \( t \) hours,
- \( a \) is the initial amount (125 words),
- \( b \) is the growth factor (2, since it doubles),
- \( t \) is the total time in hours, and
- \( h \) is the time it takes to double (8 hours).
Substituting the values into the formula, we have:
\[ y = 125 \cdot 2^{(t/8)} \]
This equation models the number of words, \( y \), the AI can learn after \( t \) hours.
To model this situation, we can use the exponential growth formula:
\[ y = a \cdot b^{(t/h)} \]
Where:
- \( y \) is the number of words after \( t \) hours,
- \( a \) is the initial amount (125 words),
- \( b \) is the growth factor (2, since it doubles),
- \( t \) is the total time in hours, and
- \( h \) is the time it takes to double (8 hours).
Substituting the values into the formula, we have:
\[ y = 125 \cdot 2^{(t/8)} \]
This equation models the number of words, \( y \), the AI can learn after \( t \) hours.
Answered by
GPT-4o mini
To determine which equation correctly models the situation where the AI program starts with 125 words and doubles its vocabulary every 8 hours, we can analyze the options.
We know that after 8 hours, the vocabulary will double from 125 to 250. This behavior indicates exponential growth. The correct model will show that the initial vocabulary is 125 words and that it doubles in the correct time frame.
Let's analyze the options based on the format:
1. **\( y = 2 \cdot (2.5)^x \)**
- This would imply a base change rate of 2.5, which is not an accurate doubling.
2. **\( y = 8 \cdot (2.5)^x \)**
- Again, this suggests a growth of 2.5 and does not represent doubling.
3. **\( y = 2 \cdot (2)^x \)**
- This suggests an increase that might suggest doubling, but starts at 2 rather than 125.
4. **\( y = 2.5 \cdot (8)^x \)**
- This would imply a much faster growth rate and doesn't fit the criteria either.
None of the options perfectly correspond to the initial value of 125 and doubling every 8 hours. However, we can analyze \( y = 2 \cdot (2)^x \) closely:
If the base is 2 and we let \( x \) represent the number of 8-hour periods, we can rewrite it considering the correct context:
- After 0 hours (\( x = 0 \)): \( y = 2^1 = 2 \),
- After 8 hours (\( x = 1 \)): \( y = 2^2 = 4 \).
This does not fit the 125 initial condition.
### Correct Exponential Model:
After 0 hours, the initial count should be \( 125 \), not \( 2 \) or \( 8 \).
The correct formula for the initial word count of 125, doubling every 8 hours, should be:
\[ y = 125 \cdot 2^{(t/8)} \]
The options don't provide the appropriate choice related to starting at 125. Thus, none of the provided options are correct for the scenario you described.
If you are looking for one of these answers closest to the characteristics of exponential growth without the initial condition being considered, **\( y = 2 \cdot (2)^x \)** follows the exponential kind but does not satisfy the initial condition correctly. This option implies a doubling growth, but starting from the wrong initial point. However, as it stands, none support the correct vocabulary of 125.
We know that after 8 hours, the vocabulary will double from 125 to 250. This behavior indicates exponential growth. The correct model will show that the initial vocabulary is 125 words and that it doubles in the correct time frame.
Let's analyze the options based on the format:
1. **\( y = 2 \cdot (2.5)^x \)**
- This would imply a base change rate of 2.5, which is not an accurate doubling.
2. **\( y = 8 \cdot (2.5)^x \)**
- Again, this suggests a growth of 2.5 and does not represent doubling.
3. **\( y = 2 \cdot (2)^x \)**
- This suggests an increase that might suggest doubling, but starts at 2 rather than 125.
4. **\( y = 2.5 \cdot (8)^x \)**
- This would imply a much faster growth rate and doesn't fit the criteria either.
None of the options perfectly correspond to the initial value of 125 and doubling every 8 hours. However, we can analyze \( y = 2 \cdot (2)^x \) closely:
If the base is 2 and we let \( x \) represent the number of 8-hour periods, we can rewrite it considering the correct context:
- After 0 hours (\( x = 0 \)): \( y = 2^1 = 2 \),
- After 8 hours (\( x = 1 \)): \( y = 2^2 = 4 \).
This does not fit the 125 initial condition.
### Correct Exponential Model:
After 0 hours, the initial count should be \( 125 \), not \( 2 \) or \( 8 \).
The correct formula for the initial word count of 125, doubling every 8 hours, should be:
\[ y = 125 \cdot 2^{(t/8)} \]
The options don't provide the appropriate choice related to starting at 125. Thus, none of the provided options are correct for the scenario you described.
If you are looking for one of these answers closest to the characteristics of exponential growth without the initial condition being considered, **\( y = 2 \cdot (2)^x \)** follows the exponential kind but does not satisfy the initial condition correctly. This option implies a doubling growth, but starting from the wrong initial point. However, as it stands, none support the correct vocabulary of 125.
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