The rate of growth of a fish population was modeled by the equation
G(t) = (60,000e−0.6t)/((1 + 5e−0.6t)^2)
where t is measured in years since 2000 and G in kilograms per year. If the biomass was 30,000 kg in the year 2000, what is the predicted biomass for the year 2020? (Round your answer to the nearest whole number.) 

To predict the biomass of the fish population for the year 2020, we need to integrate the rate of growth equation \( G(t) \) over time. The given rate of growth equation is:

\[
G(t) = \frac{60,000e^{-0.6t}}{(1 + 5e^{-0.6t})^2}
\]

where \( t \) is the number of years since 2000. In order to calculate the biomass in 2020, we'll perform the following steps:

1. **Initial Biomass at 2000**: The problem tells us that the biomass in the year 2000 (i.e., \( t = 0 \)) is 30,000 kg. 
   
2. **Years from 2000 to 2020**: We are asked to find the biomass for the year 2020, which corresponds to \( t = 20 \).

3. **Calculate the change in biomass**: To predict the biomass for the year 2020, we need to integrate \( G(t) \) over the interval \( [0, 20] \) and add it to the initial biomass.

Let's proceed with the integration and the final calculation to determine the biomass. I will calculate that for you.The predicted biomass for the fish population in the year 2020 is approximately **46,666 kilograms**.

Would you like further details on how the integration was performed or have any other questions?

Here are some related questions to further explore:
1. How does the rate of growth \( G(t) \) change over time for different values of \( t \)?
2. What would the biomass be in 2030, assuming the same model?
3. How sensitive is the biomass prediction to changes in the initial biomass?
4. What happens if we change the parameters in the growth model, like the decay rate or initial population?
5. How would a different growth model, like a logistic model, compare to this prediction?

**Tip**: When dealing with growth models, always check if the rate of growth decreases over time, indicating a stabilizing population size.

The rate of growth of a fish population was modeled by the equation G(t) = (60,000e−0.6t)/((1 + 5e−0.6t)^2) where t is measured in years since 2000 and G in kilograms per year. If the biomass was 30,000 kg in the year 2000, what is the predicted biomass for the year 2020? (Round your answer to the nearest whole number.)

Learn how to predict the biomass of a fish population in 2020 using the exponential growth model G(t) = (60,000e−0.6t)/((1 + 5e−0.6t)^2). This problem involves calculating the change in biomass from 2000 to 2020 by integrating the growth rate function, applying key calculus concepts. Suitable for advanced high school or college students.

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