A biology class at high school predictor that a local population of animals will double in size every 12 years. The population at the beginning of the year 2021 was estimated to be 50 animals. If P represents the population after n years, then which one of the following equations represents the model of the class for population?

Correct Option (D)

The problem describes exponential growth, specifically a population that doubles over a fixed period. The general formula for exponential growth with a doubling period is given by P = P₀(2)^n/d, where:

• P is the population after n years.
• P₀ is the initial population.
• 2 is the growth factor (for doubling).
• n is the number of years.
• d is the doubling period (in years).

Given that the initial population (P₀) is 50 animals and the population doubles every 12 years (d = 12), substituting these values into the formula yields:

P = 50(2)^n/12

To verify, if n = 12 years, the population should double to 100:

P = 50(2)^12/12 = 50(2)¹ = 50 × 2 = 100

This matches the condition that the population doubles every 12 years.

Incorrect Options:

• P = 12 + 50n and P = 50 + 12n:

  These equations represent linear growth models, where the population increases by a fixed amount each year, not by a doubling factor. For P = 50 + 12n, after 12 years, P = 50 + 12(12) = 50 + 144 = 194, which is not double the initial population. For P = 12 + 50n, at n=0, P=12, which contradicts the initial population of 50. Therefore, these linear models are unsuitable for describing a doubling population.

• P = 50(2)¹²ⁿ:

  This equation is an exponential model but with an incorrect exponent. If n = 12 years, P = 50(2)^12*12 = 50(2)¹⁴⁴. This value is significantly larger than 100, indicating a much faster growth rate than doubling every 12 years. The exponent 12n would imply that the population doubles 12 times per year, or every 1/12th of a year, which is contrary to the problem statement.