write the equation
3 answers
The number of words \( y \) that the AI can learn after \( x \) hours can be modeled by the following exponential equation:
\[ y = 125 \times 2^{\frac{x}{8}} \]
In this equation:
To determine which of the given equations is in the correct exponential form that models the scenario described (where the AI doubles its vocabulary every 8 hours starting from 125 words), let's briefly analyze each option.
\( y = 2.5 \cdot (8)^x \)
This equation suggests that the vocabulary grows by a factor of \( 8 \) every hour, not doubling.
\( y = 8 \cdot (2.5)^x \)
This suggests that the vocabulary grows by a factor of \( 2.5 \) every hour, which is not correct for the doubling condition.
\( y = 2 \cdot (2)^x \)
This represents \( 2 \) being multiplied by \( 2 \) every hour, not starting from \( 125 \).
\( y = 2 \cdot (2.5)^x \)
This suggests a variable growth factor of \( 2.5 \) rather than a doubling.
None of the given equations perfectly matches the exponential model. However, if we focus on the doubling of words every 8 hours, it would be best represented by our derived equation:
\[ y = 125 \cdot 2^{\frac{x}{8}} \]
Since \( 2^{\frac{x}{8}} \) indicates that vocabulary doubles (the base is 2), this is the accurate model based on the initial conditions provided.
If you wanted to convert the initial scenario into one of the forms you've provided, the only correct approach would involve adjusting them to match the doubling rate and the proper starting value.
Thus, if we strictly choose from the options you presented, none of them adequates without modifications.
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