Question

A particular Artificial Intelligence (Al) program is initially programmed to know 125 words. The Al is designed to double its vocabulary every 8 hours. Which is an exponential equation that models the number of words, y, the Al can learn after x hours? (1 point)

y = 2*(2)^x

y=2.5*(8)^x

y=8*(2.5)^x

y=2^(2.5)^x

Answer 1

To model the growth of the AI's vocabulary as it doubles every 8 hours, we start with the initial vocabulary, which is 125 words. The vocabulary doubles every 8 hours, which means that after \( x \) hours, the number of 8-hour intervals is \( \frac{x}{8} \).

The formula to model the vocabulary can be expressed as:

\[
y = \text{Initial Vocabulary} \times 2^{\text{number of intervals}}
\]

Substituting in our values:

\[
y = 125 \times 2^{\frac{x}{8}}
\]

To find a matching equation from the provided options, we can rewrite 125 as \( 5 \times 25 \) or in terms of base 2, but it is preferable to see if any option simplifies to our formula.

The equivalent exponential equation from the choices given does not exactly match our derived equation, but we can analyze them:

1. \( y = 2 \times (2)^x \) is not correct because it represents a linear growth rather than doubling every 8 hours.
2. \( y = 2.5 \times (8)^x \) does not represent the doubling behavior.
3. \( y = 8 \times (2.5)^x \) is also incorrect for the same reasons.
4. \( y = 2^{(2.5)^x} \) does not represent the modeled doubling vocabulary based on time.

None of the provided options accurately reflects the equation derived for this scenario.

If rephrasing or adapting the options is an option, it would be better to depict any answer aligning with \( y = 125 \times 2^{\frac{x}{8}} \).

None of the statements provided are correct options for the exponential equation modeling the vocabulary increase of the AI after \( x \) hours based on the rules given.