Question

a particular artificial intelligence (ai) program is initially programmed to know 125 words. the ai is designed to double its vocabulary every 8 hours at the same rate. which is an exponential equation that models the number of words, y, the ai can learn after x sets of 8 hours? (1 point)
\\( y = 125 \cdot (8)^x \\)
\\( y = 2 \cdot (125)^x \\)
\\( y = 250 \cdot (2)^x \\)
\\( y = 125 \cdot (2)^x \\)
graphing calculator

Explanation:

Step1: Recall exponential growth formula

The general exponential growth formula is $y = a \cdot b^x$, where $a$ is the initial amount, $b$ is the growth factor, and $x$ is the number of time periods.

Step2: Identify initial value $a$

The AI starts with 125 words, so $a = 125$.

Step3: Identify growth factor $b$

The vocabulary doubles every 8-hour period, so the growth factor $b = 2$.

Step4: Substitute values into formula

Substitute $a=125$ and $b=2$ into $y = a \cdot b^x$, giving $y = 125 \cdot (2)^x$.

Answer:

$y = 125 \cdot (2)^x$