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
Answers
GPT-4o mini
To model the number of words the AI can learn after \( x \) hours, we can use an exponential growth formula. Given that the AI starts with 125 words and doubles its vocabulary every 8 hours, the formula for the vocabulary at time \( x \) can be expressed as follows:
\[
W(x) = W_0 \cdot 2^{\frac{x}{d}}
\]
where:
- \( W(x) \) is the number of words the AI knows after \( x \) hours,
- \( W_0 \) is the initial number of words (125 words),
- \( d \) is the doubling time in hours (8 hours),
- \( x \) is the number of hours passed.
Substituting the values into the equation, we get:
\[
W(x) = 125 \cdot 2^{\frac{x}{8}}
\]
This equation calculates the number of words the AI can learn after \( x \) hours. For example, after 8 hours, it would be:
\[
W(8) = 125 \cdot 2^{\frac{8}{8}} = 125 \cdot 2^1 = 125 \cdot 2 = 250
\]
Thus, after 8 hours, the AI would know 250 words. After 16 hours, it would know:
\[
W(16) = 125 \cdot 2^{\frac{16}{8}} = 125 \cdot 2^2 = 125 \cdot 4 = 500
\]
You can use this formula to determine the vocabulary size at any given number of hours \( x \).
\[
W(x) = W_0 \cdot 2^{\frac{x}{d}}
\]
where:
- \( W(x) \) is the number of words the AI knows after \( x \) hours,
- \( W_0 \) is the initial number of words (125 words),
- \( d \) is the doubling time in hours (8 hours),
- \( x \) is the number of hours passed.
Substituting the values into the equation, we get:
\[
W(x) = 125 \cdot 2^{\frac{x}{8}}
\]
This equation calculates the number of words the AI can learn after \( x \) hours. For example, after 8 hours, it would be:
\[
W(8) = 125 \cdot 2^{\frac{8}{8}} = 125 \cdot 2^1 = 125 \cdot 2 = 250
\]
Thus, after 8 hours, the AI would know 250 words. After 16 hours, it would know:
\[
W(16) = 125 \cdot 2^{\frac{16}{8}} = 125 \cdot 2^2 = 125 \cdot 4 = 500
\]
You can use this formula to determine the vocabulary size at any given number of hours \( x \).