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

Question

Responses

y=2⋅(2)x
𝑦
=
2
⋅
(
2
)
𝑥
y is equal to 2 times 2 to the x th power

y=8⋅(2.5)x
𝑦
=
8
⋅
(
2.5
)
𝑥
y is equal to 8 times 2 point 5 to the x th power

y=2.5⋅(8)x
𝑦
=
2.5
⋅
(
8
)
𝑥
y is equal to 2 point 5 times 8 to the x th power

y=2⋅(2.5)x

Answer 1

To model the vocabulary growth of the AI over time, we can use the knowledge that the AI starts with 125 words and doubles its vocabulary every 8 hours.

Let’s break down the components:

1. Starting Vocabulary: The AI starts with 125 words.
2. Doubling Time: The AI doubles its vocabulary every 8 hours.

Since the vocabulary doubles every 8 hours, we can express this with an exponential function.

After \( x \) hours, the number of 8-hour intervals is \( \frac{x}{8} \).

The general formula for exponential growth is:

\[ y = y_0 \cdot 2^{(t/T)} \]

where:

• \( y_0 \) is the initial amount,
• \( t \) is the total time passed,
• \( T \) is the time it takes to double.

Substituting in the values:

• \( y_0 = 125 \),
• \( t = x \),
• \( T = 8 \),

The equation becomes:

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

Now, we can express this more clearly by rewriting \( 125 \) as \( 5^3 \times 2^0 \) (since \( 125 = 5^3 \)). However, the given options do not contain a formulation starting with this value.

The equation \( y = 125 \cdot 2^{\frac{x}{8}} \) does not fit any of the provided options.

Among the choices provided:

• None of the options given are a correct match for the exponential model based on the parameters specified.

If I had to rewrite the exponent to match one of the choices, I could rewrite \( 125 \) (as \( 2^7.3219 \approx 125 \)) and then match it against the doubling rate; however, that will be very approximate.

Therefore, please clarify any constraints regarding the vocabulary growth function that might match one of the provided options, or if those options are to be reconsidered based on the base growth model.

Answer 2

you have to choose one

Answer 3

Given the options presented, the closest option that reflects an exponential growth model (despite not perfectly fitting the criteria) is:

y = 2 ⋅ (2)^x

This equation can be interpreted with some adjustments. In relation to vocabulary doubling every 8 hours, this might represent a simplified quantity of words that do not align perfectly with the starting vocabulary of 125 words but reflects the exponential growth pattern of doubling.

Therefore, I would choose:

y = 2 ⋅ (2)^x

If we're strictly limited to choosing one of the provided options, this would be it.