A biology class at Central High School predicted that a local population of animals will double in size every 12 years. The population at the beginning of 2021 was estimated to be 50 animals. If P represents the population n years after 2021, then which of the following equations represents the class model of the population over time?

The problem describes exponential growth, where the population doubles every 12 years. The initial population is 50.

1. Identify the type of growth: The phrase "doubles in size every 12 years" indicates exponential growth, not linear growth.
2. General form of exponential growth: For a population that doubles every 'd' units of time, the formula is P = P0 * (2)^(t/d), where P0 is the initial population, t is the time elapsed, and d is the doubling period.
3. Apply the given values:
   • Initial population (P0) = 50
   • Doubling period (d) = 12 years
   • Time elapsed (t) = n years
4. Substitute these values into the formula: P = 50 * (2)^(n/12).

Let's evaluate the options:

A) P = 12 + 50n: This represents linear growth. If n=0, P=12, which is incorrect as the initial population is 50. B) P = 50 + 12n: This also represents linear growth. If n=12, P = 50 + 1212 = 194. However, after 12 years, the population should double to 502 = 100. So, this is incorrect. C) P = 50(2)^12n: This is an exponential model, but the exponent is incorrect. If n=12, P = 50(2)^(1212) = 50(2)^144, which is an extremely large number, not 100. This model would imply doubling 12 times per year, or a much faster growth rate. D) P = 50(2)^(n/12): This matches the derived exponential growth formula. * If n=0 (beginning of 2021), P = 50(2)^(0/12) = 50(2)^0 = 501 = 50. (Correct initial population) * If n=12 (after 12 years), P = 50(2)^(12/12) = 50(2)^1 = 50*2 = 100. (Correct doubling)

Therefore, option D correctly represents the class model of the population over time.

The final answer is D