Questions: A species of fish was added to a lake. The population size P(t) of this species can be modeled by the time the species was added to the lake. P(t) = 1200 / (1 + 7 e^(-0.1 t)) Find the population size of the species after 3 years and the population size after 7 years. Round your answers to the nearest whole number as necessary. Population size after 3 years: fish Population size after 7 years: fish

A species of fish was added to a lake. The population size P(t) of this species can be modeled by the time the species was added to the lake.

P(t) = 1200 / (1 + 7 e^(-0.1 t))

Find the population size of the species after 3 years and the population size after 7 years. Round your answers to the nearest whole number as necessary.

Population size after 3 years: fish

Population size after 7 years: fish

Transcript text: A species of fish was added to a lake. The population size $P(t)$ of this species can be modeled by the time the species was added to the lake. \[ P(t)=\frac{1200}{1+7 e^{-0.1 t}} \] Find the population size of the species after 3 years and the population size after 7 years. Round your answers to the nearest whole number as necessary. Population size after 3 years: $\square$ fish Population size after 7 years: $\square$ fish

Solution

Solution Steps

To solve this problem, we need to evaluate the given exponential function $ P(t) = \frac{1200}{1 + 7e^{-0.1t}} $ at $ t = 3 $ and $ t = 7 $. This will give us the population size of the species after 3 years and 7 years, respectively. We will use Python to perform these calculations and round the results to the nearest whole number.

Solution Approach

Define the function $ P(t) $ in Python.
Evaluate the function at $ t = 3 $ and $ t = 7 $.
Round the results to the nearest whole number.

Step 1: Define the Population Model

The population size $ P(t) $ of the species of fish in the lake is modeled by the function: \[ P(t) = \frac{1200}{1 + 7 e^{-0.1 t}} \]

Step 2: Calculate Population After 3 Years

To find the population size after 3 years, we substitute $ t = 3 $ into the function: \[ P(3) = \frac{1200}{1 + 7 e^{-0.1 \cdot 3}} \approx 194 \] Thus, the population size after 3 years is approximately 194 fish.

Step 3: Calculate Population After 7 Years

Next, we calculate the population size after 7 years by substituting $ t = 7 $: \[ P(7) = \frac{1200}{1 + 7 e^{-0.1 \cdot 7}} \approx 268 \] Therefore, the population size after 7 years is approximately 268 fish.