To model the number of bacteria \( y \) in terms of time \( x \), we need to observe the pattern in the given data points:
- When \( x = 0 \), \( y = 192 \)
- When \( x = 1 \), \( y = 96 \)
- When \( x = 2 \), \( y = 48 \)
- When \( x = 3 \), \( y = 24 \)
- When \( x = 4 \), \( y = 12 \)
Looking at these values, it appears each time step results in the bacteria count being halved. This suggests an exponential decay model can be used.
The general form for exponential decay can be expressed as:
\[ y = y_0 \cdot a^x \]
Where:
- \( y_0 \) is the initial amount of bacteria.
- \( a \) is the decay factor (in this case, it would be \( \frac{1}{2} \) since the population halves every second).
- \( x \) is the time in seconds.
From the data, the initial amount of bacteria \( y_0 = 192 \) when \( x = 0 \).
Therefore, the function modeling the number of bacteria over time would be:
\[ y = 192 \left( \frac{1}{2} \right)^x \]
This equation accurately describes the behavior of the bacteria count as it halves every second.
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