Final answer:
To find the rabbit population after 8 years with a 25% growth rate every 3 years, you use exponential growth calculations over two full 3-year periods and then adjust for the remaining 2 years, resulting in an estimated population of approximately 34,078 rabbits.
Step-by-step explanation:
To calculate the population of rabbits after 8 years, given a growth rate of 25% every 3 years, we can use the formula for exponential growth. The formula is given by P = P0 (1 + r)n, where P is the final population, P0 is the initial population, r is the growth rate per period, and n is the number of periods.
In this scenario, the initial population (P0) is 20,000 rabbits, the growth rate (r) is 0.25 (or 25% expressed as a decimal), and the periods (n) are calculated by dividing the time in years by the number of years per period, which is 8 years ÷ 3 years/period = approximately 2.67 periods. However, since the population grows only at the end of each period, we will consider growth over 2 full periods (6 years) and then calculate the additional two years separately.
After 6 years (2 periods):
P6 = 20,000 x (1 + 0.25)2 = 20,000 x 1.252 = 31,250
For the remaining 2 years, we apply a growth rate for two-thirds of a three-year period. Since this is not a full period, we need to adjust the growth rate proportionally. A 25% increase over 3 years can be converted to an annual rate by taking the cube root of 1.25, which gives us approximately 1.0801. The growth for two-thirds of a year would be 1.08012/3.
After the additional 2 years:
P8 = P6 x 1.08012/3 = 31,250 x 1.08012/3 = 34,078.47, approximately 34,078 rabbits.
Therefore, the estimated rabbit population after 8 years will be approximately 34,078 rabbits.