To create an exponential decay function to model the decrease in the population of birds, we can use the formula:
\[ P(t) = P_0 \cdot e^{kt} \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant (as a negative value for decay),
- \( t \) is time in years,
- \( e \) is the base of the natural logarithm.
Given:
- The initial population \( P_0 = 800 \) birds,
- The rate of decrease is 4% per year, which translates to \( k = -0.04 \).
So, the function becomes:
\[ P(t) = 800 \cdot e^{-0.04t} \]
To find the population after 15 years (\( t = 15 \)):
\[ P(15) = 800 \cdot e^{-0.04 \cdot 15} \]
Calculating the exponent:
\[ -0.04 \cdot 15 = -0.6 \]
Now substituting this value into the function:
\[ P(15) = 800 \cdot e^{-0.6} \]
Using the approximate value of \( e^{-0.6} \approx 0.5488 \):
\[ P(15) \approx 800 \cdot 0.5488 \]
Calculating the final population:
\[ P(15) \approx 439.04 \]
Rounding to the nearest whole number, the estimated number of birds in the forest after 15 years is:
439 birds.
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