To model the growth of the AI's vocabulary, we can use the formula for exponential growth, which has the general form:
\[ y = a \cdot b^{(t / T)} \]
where:
- \( y \) is the final quantity (number of words),
- \( a \) is the initial quantity (initial number of words),
- \( b \) is the growth factor,
- \( t \) is the elapsed time,
- \( T \) is the time it takes to double.
Given that the AI initially knows 125 words (so \( a = 125 \)) and it doubles its vocabulary every 8 hours (so every 8 hours \( b = 2 \)), the equation that models the number of words after \( x \) hours can be expressed as:
\[ y = 125 \cdot 2^{(x / 8)} \]
However, since this specific form is not among the options given, we can convert it into one of the choices.
To get a clearer look, find how the initial quantity of 125 can be expressed in terms of smaller bases:
- \( 125 = 5^3 \)
Then we can rewrite our model slightly.
Now looking at the choices provided, we can analyze the exponential growth.
Among the choices:
- y = 2.5 ⋅ (8)^x - This has a base of 8 but not matching the doubling condition.
- y = 2 ⋅ (2)^x - Incorrect, as it suggests constant doubling every hour rather than every 8 hours.
- y = 2 ⋅ (2.5)^x - The base isn’t precise for this scenario.
- y = 8 ⋅ (2.5)^x - The 2.5 is not an exact fit for exponential doubling.
Given that none of the options perfectly express the doubling vocabulary model correctly after hours, there’s no exact answer choice fitting the correctly derived formula \( y = 125 \cdot 2^{(x / 8)} \) based on the condition.
That said, focusing on a structured approximation that fits as close as possible leads to agreements with logical examination and choice elimination aligning primarily around the factor base and doubling timelines where the closest that matches the initializing point of view would typically resonate.
Therefore, based on the exponential growth logic and given parameters to search through, you might arrive at the conclusion that none of the provided options perfectly match. If you would have to choose the one most aligned, it might be necessary to work with \( y = 2.5 ⋅ (8)^x \) under specific context altering interpretations or correct initial function adjustments.