To create an exponential decay function to model the population of the birds, we can use the formula for exponential decay, which 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 years,
- \(e\) is the base of the natural logarithm.
Since the population is decreasing at a rate of 4% per year, we can express \(k\) as:
\[ k = -0.04 \]
Given that the current population \(P_0\) is 800, we can substitute the values into the formula:
\[ P(t) = 800 \cdot e^{-0.04t} \]
Now, we need to find the population after 15 years. Therefore, we will substitute \(t = 15\) into the equation:
\[ P(15) = 800 \cdot e^{-0.04 \cdot 15} \]
Calculating the exponent:
\[ -0.04 \cdot 15 = -0.6 \]
Now substitute this back into the equation:
\[ P(15) = 800 \cdot e^{-0.6} \]
Next, we calculate \(e^{-0.6}\). Using a calculator, we find:
\[ e^{-0.6} \approx 0.5488 \]
Now substituting this value back into the equation gives us:
\[ P(15) \approx 800 \cdot 0.5488 \approx 439.04 \]
Rounding this to the nearest whole number, we find:
\[ P(15) \approx 439 \]
Thus, after 15 years, the estimated population of the birds in the forest will be approximately 439 birds.
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