Questions: A species of animal is discovered on an island. Suppose that the population size P(t) of the species can be modeled by the following function, where time t is measured in years. P(t) = 800 / (1 + 2 e^(-0.45 t)) Find the population size after 6 years and after 9 years. Round your answers to the nearest whole number as necessary.

A species of animal is discovered on an island. Suppose that the population size P(t) of the species can be modeled by the following function, where time t is measured in years. P(t) = 800 / (1 + 2 e^(-0.45 t)) Find the population size after 6 years and after 9 years. Round your answers to the nearest whole number as necessary.
Transcript text: A species of animal is discovered on an island. Suppose that the population size $P(t)$ of the species can be modeled by the following function, where time $t$ is measured in years. \[ P(t)=\frac{800}{1+2 e^{-0.45 t}} \] Find the population size after 6 years and after 9 years. Round your answers to the nearest whole number as necessary.
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Solution

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Solution Steps

Step 1: Calculate Population Size After 6 Years

To find the population size after 6 years, we evaluate the function \( P(t) \) at \( t = 6 \): \[ P(6) = \frac{800}{1 + 2 e^{-0.45 \cdot 6}} \approx 705 \]

Step 2: Calculate Population Size After 9 Years

Next, we evaluate the function \( P(t) \) at \( t = 9 \): \[ P(9) = \frac{800}{1 + 2 e^{-0.45 \cdot 9}} \approx 773 \]

Final Answer

The population size after 6 years is approximately \( \boxed{705} \) and after 9 years is approximately \( \boxed{773} \).

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