Understanding Population Dynamics in a Fish Pond: Correcting the Equation
Fish populations in a pond are often modeled through mathematical equations to predict their growth or decline. In a scenario where you start with 100 fish, and the population changes at a rate of 4t3-12t-150 where t is time in weeks, the initial problem might seem straightforward. However, an incorrect interpretation of the initial conditions can lead to significant misunderstandings. This article will explain how to adjust the equation to reflect the initial population accurately.
Initial Conditions and Mathematical Modeling
In mathematical modeling, initial conditions are crucial. They provide the starting point from which the model will evolve. In the context of the fish pond, the initial condition is that there are initially 100 fish. This means that at t0, the population should be 100. However, if we directly substitute t0 into the equation, we get a nonsensical result: 4(0)3-12(0)-150 -150. This negative value doesn't make sense in the context of the fish population.
Correcting the Equation
To correct the equation and ensure it reflects the initial population, we need to adjust the function. The correct equation should be:
Population change rate 4t3 - 12t 100
This adjustment ensures that at t0, we have:
4(0)3 - 12(0) 100 100
With this corrected equation, we can now calculate the population of fish after one year (52 weeks). Using the corrected equation, the integral of the rate of change will give us the total change in population over time.
Calculating the Population After 1 Year
The population change over time can be calculated by integrating the rate of change function. The integral of 4t3 - 12t 100 is:
∫(4t3 - 12t 100)dt t4/1 - 6t2 100t
Evaluating this integral from 0 to 52 weeks:
[t4/1 - 6t2 100t] from 0 to 52
[(524/1 - 6*522 100*52) - (04/1 - 6*02 100*0)]
56,190,880 - 0 56,190,880
Adding the initial 100 fish to the computed change:
Final population 100 56,190,880 56,190,980
Conclusion
Therefore, after 52 weeks (one year), the population of fish in the pond will be approximately 56,190,980. This correct interpretation and application of the initial conditions allow us to predict the population accurately, far from the intuitive result of all fish being dead in just one week.