To model the decline of the ant population, we can use the exponential decay formula:
\[ y = y_0 \cdot (1 - r)^x \]
Where:
- \(y\) is the population after \(x\) weeks,
- \(y_0\) is the initial population,
- \(r\) is the rate of decline (as a decimal),
- \(x\) is the number of weeks.
Given:
- The initial population \(y_0 = 250,000\),
- The rate of decline \(r = 2.5% = 0.025\).
Substituting these values into the formula, we have:
\[ y = 250,000 \cdot (1 - 0.025)^x \]
This simplifies to:
\[ y = 250,000 \cdot (0.975)^x \]
This is the exponential equation that models the ant population after \(x\) weeks.
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