To model the number of words, \(y\), that the AI can learn after \(x\) hours, we start with the information that the AI starts with 125 words and doubles its vocabulary every 8 hours.
The general form of an exponential growth model is:
\[ y = a \cdot b^{(x/t)} \]
where:
- \(a\) is the initial amount of the population (in this case, the initial number of words, which is 125).
- \(b\) is the growth factor (since the vocabulary doubles, \(b = 2\)).
- \(t\) is the time it takes to double (8 hours).
- \(x\) is the time passed.
Given the conditions, the equation becomes:
\[ y = 125 \cdot 2^{(x/8)} \]
None of the provided responses perfectly match this equation. However, we can manipulate the equation to match the form better:
Let's express 125 in terms of powers of 2. We know:
\[ 125 = 5^3 \]
This does not provide a neat power of 2, however, for precise matching and simplification, we can work directly with the doubling.
If we were to express the equation in simpler base terms, we recognize that the factor of growth is essentially based on the doubling provided every 8 hours.
If we keep our analysis based on the principle of words doubling every 8 hours, we can describe the number of words more generally with respect to its growth factor (2) over the intervals of time (8 hours).
Thus:
- For every 8 hours, the number of words doubles, so when \(x = 8\), \(y = 125 \cdot 2\).
- When \(x = 16\), \(y = 125 \cdot 2^2\), and so on.
This reflects exponential growth at any interval of 8 until extrapolating to any given hour \(x\).
Overall, given the question provides options without a clear match, the answer can be best approximated logically although not listed:
Correct form: \[ y = 125 \cdot 2^{(x/8)} \]
Best alternative option based on description of responses: The option that expresses a base or doubling principle can sometimes roughly fit but will need to be matched to the practical growing model explained above.
In conclusion, the equation is ultimately guided primarily by the condition of doubling every 8 hours, meaning:
The provided options seem not to match perfectly, but rephrasing or further vectors in the models might help draw the best insight. If needing simplicity, this doubling mechanism over hours is critical in understanding the AI vocabulary growth.