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On day one, the population of gray wolves in Montana was 2900 wolves, and 6 years later the population was 3700 wolves.

a) Find the function \( P(t) \) which gives the population of wolves \( t \) years later.

Answer :

The function P(t) that gives the population of gray wolves t years later is:

P(t) = 2900 * e^(133.33t).

To find the function P(t) that gives the population of wolves t years later, we can use the given information to determine the growth rate of the wolf population.

Let's define the initial population of wolves as P₀ = 2900 and the population after 6 years as P₆ = 3700.

The growth rate can be calculated by dividing the change in population by the number of years:

Growth rate = (Final population - Initial population) / Number of years

Substituting the given values, we have:

Growth rate = (3700 - 2900) / 6

Growth rate = 800 / 6

Growth rate = 133.33 (rounded to two decimal places)

Now that we have the growth rate, we can use it to determine the function P(t). The general form of exponential growth is given by the equation:

P(t) = P₀ * e^(rt)

Where P(t) is the population at time t, P₀ is the initial population, r is the growth rate, and e is the base of the natural logarithm.

Substituting the given values, we have:

P(t) = 2900 * e^(133.33t)

Therefore, the function P(t) that gives the population of gray wolves t years later is:

P(t) = 2900 * e^(133.33t).

This function describes the exponential growth of the wolf population in Montana over time.

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