To model the population decline of the ant colony using an exponential equation, we can start with the general formula for exponential decay:
\[ P(t) = P_0 \times e^{-kt} \]
Where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population,
- \(k\) is the decay constant, and
- \(t\) is the time in weeks.
Given that the initial population \(P_0\) is 250,000 ants and the population declines at a rate of 2.5% per week, we first need to convert this percentage into the decay constant \(k\).
The relationship between the rate of decline \(r\) (in decimal form) and the decay constant \(k\) is:
\[ k = -\ln(1 - r) \]
For a 2.5% decline, we can express \(r\) as:
\[ r = 0.025 \]
Now we can calculate \(k\):
\[ k = -\ln(1 - 0.025) \approx -\ln(0.975) \approx 0.0253 \]
Thus, our exponential decay model becomes:
\[ P(t) = 250,000 \times e^{-0.0253t} \]
Alternatively, you can express the model without using \(e\), as:
\[ P(t) = 250,000 \times (1 - 0.025)^t = 250,000 \times (0.975)^t \]
Therefore, the exponential equation to model the ant population after \(t\) weeks is:
\[ P(t) = 250,000 \times (0.975)^t \]
This equation describes the population of ants at any given week \(t\).