Final answer:
To find the critical point for achieving the desired population growth of rabbits introduced in a park, we can use the exponential growth model. By plugging in the growth rate and initial population into the formula, we can solve for the time it takes for the population to reach the desired growth. The critical point is when the population is approximately 5,010 rabbits.
Step-by-step explanation:
To determine the critical point for achieving the desired population growth, we need to use the exponential growth model. The formula for exponential growth is P(t) = P0 * e^(rt), where P(t) is the population size at time t, P0 is the initial population size, e is the base of the natural logarithm, and r is the growth rate.
In this case, we are given that the population will grow at a rate of 80 rabbits per year. So r = 80. The initial population is 5,000, so P0 = 5,000.
Substituting these values into the formula, we have:
P(t) = 5,000 * e^(80t)
To find the critical point, we need to solve for t when P(t) reaches the desired population growth. Since the question asks for the answer rounded to the nearest whole number, we can use logarithms to solve for t:
e^(80t) = 5,000 / P0
80t = ln(5,000 / P0)
t = ln(5,000 / P0) / 80
Plugging in the values for P0 and rounding the result to the nearest whole number, we get t ≈ 5. Thus, the critical point for these rabbits to achieve the desired population growth would be when the population is approximately 5,010 rabbits, which corresponds to option a).