Question

And a certain Forest the population of a particular species of birds is decreasing at a rate of 4% per year if there are currently 800 birds in the forest create an exponential decay function to model the problem if the decrease in population continues at this rate how many birds will be in the forest after 15 years round your answer to the nearest whole number

Answer 1

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.**