To model the population of a colony of ants that initially has 25,000 ants and declines at a rate of 2.5% each week, we can use the formula for exponential decay.
The general formula for exponential decay is given by:
\[ P(t) = P_0 \cdot e^{-kt} \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant,
- \( t \) is the time in weeks,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
However, since we have a percentage decline, we can express the population model in a simpler form:
\[ P(t) = P_0 \cdot (1 - r)^t \]
where \( r \) is the decay rate expressed as a decimal.
Given:
- Initial population \( P_0 = 25,000 \),
- Rate of decline \( r = 2.5% = 0.025 \).
Thus, the equation becomes:
\[ P(t) = 25000 \cdot (1 - 0.025)^t \] \[ P(t) = 25000 \cdot (0.975)^t \]
This equation will model the population of the ant colony after \( t \) weeks.
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