To model the ant population \( y \) after \( x \) weeks, given that the population declines at a rate of 2.5% each week, we can use the formula for exponential decay:
\[ y = y_0 \cdot (1 - r)^x \]
where:
- \( y_0 \) is the initial population,
- \( r \) is the decay rate (in decimal form),
- \( x \) is the number of weeks,
- \( y \) is the population after \( x \) weeks.
Given:
- \( y_0 = 250,000 \)
- \( r = 0.025 \) (which is 2.5% expressed as a decimal)
Therefore, the equation becomes:
\[ y = 250000 \cdot (1 - 0.025)^x \]
Simplifying \( (1 - 0.025) \):
\[ y = 250000 \cdot (0.975)^x \]
So the exponential equation to model the insect population after \( x \) weeks is:
\[ y = 250000 \cdot (0.975)^x \]
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