To model the number of bacteria \( y \) in terms of time \( x \), we can observe the provided data points:
- At \( x = 0 \), \( y = 200 \)
- At \( x = 1 \), \( y = 50 \)
- At \( x = 2 \), \( y = 12.5 \)
- At \( x = 3 \), \( y = 3.125 \)
Notice that as time \( x \) increases, the number of bacteria \( y \) decreases significantly. It appears to decrease by a factor of 4 with each passing second:
- \( 200 \div 4 = 50 \)
- \( 50 \div 4 = 12.5 \)
- \( 12.5 \div 4 = 3.125 \)
This suggests an exponential decay model. Given this pattern, we can express the relationship between \( y \) and \( x \) using the following equation:
\[ y = 200 \cdot \left( \frac{1}{4} \right)^x \]
This function indicates that the number of bacteria decreases to one-quarter of its previous value with each passing second. To confirm:
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For \( x = 0 \): \[ y = 200 \cdot \left( \frac{1}{4} \right)^0 = 200 \cdot 1 = 200 \]
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For \( x = 1 \): \[ y = 200 \cdot \left( \frac{1}{4} \right)^1 = 200 \cdot \frac{1}{4} = 50 \]
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For \( x = 2 \): \[ y = 200 \cdot \left( \frac{1}{4} \right)^2 = 200 \cdot \frac{1}{16} = 12.5 \]
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For \( x = 3 \): \[ y = 200 \cdot \left( \frac{1}{4} \right)^3 = 200 \cdot \frac{1}{64} = 3.125 \]
Thus, the final function modeling the number of bacteria in terms of time is:
\[ y = 200 \cdot \left( \frac{1}{4} \right)^x \]