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:
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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.
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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.