Suppose a population $ P $ of bacteria satisfies the hypothesis of the exponential population model: The rate of change of the population is proportional to the population. Suppose the initial population is $ 1000 $ and the population after 3 hours is $ 60000 $.

1. Establish and resolve an Initial Value Problem to express population $ P $ as a function of time $ t $ in hours.

2. Find the population after 25 hours.

Express the answer in an explicit form $ y = f(x) $.

To find the population after 25 hours using the exponential population model, we first need to determine the value of the proportionality constant, k. Let's proceed with the calculations.

From the previous explanation, we have the equation ln(60) = 3k. Solving this equation for k, we find:

k = ln(60) / 3 ≈ 0.609

Now that we have the value of k, we can use it to calculate the population after 25 hours. Plugging in the values into the exponential function, we have:

P(25) = 1000 * [tex]e^{(0.609 * 25)[/tex]

Using a calculator or software, we can evaluate this expression to find:

P(25) ≈ 26925.37

Therefore, the population after 25 hours, according to the given exponential model, is approximately 26925.37.

The exponential population model is a common mathematical approach to describe population growth or decay in various fields.

It assumes that the rate of change of the population is proportional to the population itself.

Solving the initial value problem allows us to determine the specific exponential function that represents the population as a function of time.

By finding the values of P0 and k, we can express the population function explicitly.

It is important to note that the exponential model is a simplification and may not accurately capture all factors influencing population dynamics in real-world scenarios.

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