To model the number of bacteria \( y \) in terms of time \( x \) based on the data provided, we can observe the decreasing pattern of the bacteria count.
The data indicates:
- At \( x = 0 \), \( y = 100 \)
- At \( x = 1 \), \( y = 50 \)
- At \( x = 2 \), \( y = 25 \)
- At \( x = 3 \), \( y = 12.5 \)
- At \( x = 4 \), \( y = 6.25 \)
This data suggests that the number of bacteria is halved every second. Therefore, this represents an exponential decay model.
A general form of the exponential decay function is:
\[ y = y_0 \cdot a^x \]
where:
- \( y_0 \) is the initial quantity (at \( x = 0 \)),
- \( a \) is the decay factor,
- \( x \) is time.
From the data, the initial number of bacteria \( y_0 = 100 \).
Since the number of bacteria is half every second, the decay factor \( a = \frac{1}{2} \).
Thus, the equation modeling the number of bacteria over time \( x \) can be written as:
\[ y = 100 \left( \frac{1}{2} \right)^x \]
This function describes the exponential decay of the bacteria count over time.
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