In 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.

Question

simple math

Answer 1

To model the population of the birds using an exponential decay function, we can use the formula:

\[ P(t) = P_0 \cdot (1 - r)^t \]

where:

• \( P(t) \) is the population at time \( t \),
• \( P_0 \) is the initial population,
• \( r \) is the decay rate (as a decimal),
• \( t \) is the time in years.

In this case:

• \( P_0 = 800 \) (the current population),
• \( r = 0.04 \) (the decay rate of 4%),
• \( t = 15 \) (the number of years).

Substituting these values into the formula gives:

\[ P(15) = 800 \cdot (1 - 0.04)^{15} \]

This simplifies to:

\[ P(15) = 800 \cdot (0.96)^{15} \]

Now, we can calculate \( (0.96)^{15} \):

\[ (0.96)^{15} \approx 0.442(4) , (\text{using a calculator}) \]

So now substituting this back in:

\[ P(15) = 800 \cdot 0.442 \approx 353.6 \]

Rounding to the nearest whole number, the estimated population after 15 years will be:

\[ \approx 354 \text{ birds} \]

Thus, there will be approximately 354 birds in the forest after 15 years.

Answer 2

In a specific lake, the fish population is declining at a rate of 3.5% per year. If there are currently 1,200 fish in the lake, create an exponential decay function to model the problem. If the decrease in population continues at this rate, how many fish will be in the lake after 20 years? Round your answer to the nearest whole number.
simple answer

Answer 3

To model the fish population using an exponential decay function, we use the formula:

\[ P(t) = P_0 \cdot (1 - r)^t \]

where:

• \( P(t) \) is the population at time \( t \),
• \( P_0 \) is the initial population,
• \( r \) is the decay rate (as a decimal),
• \( t \) is the time in years.

For this problem:

• \( P_0 = 1200 \),
• \( r = 0.035 \) (3.5% expressed as a decimal),
• \( t = 20 \).

Substituting these values into the formula:

\[ P(20) = 1200 \cdot (1 - 0.035)^{20} \]

This simplifies to:

\[ P(20) = 1200 \cdot (0.965)^{20} \]

Next, we calculate \( (0.965)^{20} \):

\[ (0.965)^{20} \approx 0.509 (using a calculator) \]

Now substituting this back in:

\[ P(20) = 1200 \cdot 0.509 \approx 611.4 \]

Rounding to the nearest whole number, the estimated fish population after 20 years will be:

\[ \approx 611 \text{ fish} \]

So, there will be approximately 611 fish in the lake after 20 years.