Question
Particular AI program is initially programmed to know 125 words that AI does is designed to double its vocabulary every 8 hours which is an exponential equation that models the number of words why the AI can learn after x hours
Y = 8 * 2.5 x y = 2 * 2x = 2 * 2.5 XY = 2.5 * 8x
Y = 8 * 2.5 x y = 2 * 2x = 2 * 2.5 XY = 2.5 * 8x
Answers
GPT-4o mini
To model the growth of the AI's vocabulary over time, let's clarify the parameters given in your problem:
1. The AI starts with an initial vocabulary of 125 words.
2. The AI's vocabulary doubles every 8 hours.
This can be expressed as an exponential growth model. The general form for exponential growth can be written as:
\[ Y = Y_0 \cdot a^{\frac{x}{b}} \]
Where:
- \( Y \) is the number of words after \( x \) hours.
- \( Y_0 \) is the initial amount (125 words).
- \( a \) is the growth factor (in this case, \( 2 \), since the vocabulary doubles).
- \( b \) is the time period over which the growth occurs (8 hours).
Plugging these values into the equation gives us:
\[ Y = 125 \cdot 2^{\frac{x}{8}} \]
In this representation:
- \( x \) is the number of hours that have passed.
- \( Y \) is the total number of words the AI knows after \( x \) hours.
### Summary Equation
So, the correct exponential equation that models the number of words the AI can learn after \( x \) hours is:
\[ Y = 125 \cdot 2^{\frac{x}{8}} \]
This equation accurately reflects the starting vocabulary of 125 words that doubles every 8 hours.
1. The AI starts with an initial vocabulary of 125 words.
2. The AI's vocabulary doubles every 8 hours.
This can be expressed as an exponential growth model. The general form for exponential growth can be written as:
\[ Y = Y_0 \cdot a^{\frac{x}{b}} \]
Where:
- \( Y \) is the number of words after \( x \) hours.
- \( Y_0 \) is the initial amount (125 words).
- \( a \) is the growth factor (in this case, \( 2 \), since the vocabulary doubles).
- \( b \) is the time period over which the growth occurs (8 hours).
Plugging these values into the equation gives us:
\[ Y = 125 \cdot 2^{\frac{x}{8}} \]
In this representation:
- \( x \) is the number of hours that have passed.
- \( Y \) is the total number of words the AI knows after \( x \) hours.
### Summary Equation
So, the correct exponential equation that models the number of words the AI can learn after \( x \) hours is:
\[ Y = 125 \cdot 2^{\frac{x}{8}} \]
This equation accurately reflects the starting vocabulary of 125 words that doubles every 8 hours.