A biology class at high school predicted 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 the population ?
- (a) P = 12 + 50n
- (b) P = 50 + 12n
- (c) P = 50 (2)^12n
- (d) P = 50 (2)^n/12 ✓ UPSC's answer
Why the answer is (d)
• The initial population in 2021 is given as 50, which serves as the base value in the exponential growth model.
• The population doubles every 12 years, meaning the growth factor is 2 raised to the power of the number of 12-year periods elapsed.
• If $n$ represents the total number of years, the number of 12-year periods is calculated as $n/12$.
• Therefore, the population $P$ after $n$ years is the initial population multiplied by $2$ raised to the power of $n/12$.
• This yields the equation $P = 50(2)^{n/12}$, which matches option (d).
Why the other options are wrong
- (a) P = 12 + 50n
- This equation represents linear growth with a slope of 50 and an intercept of 12, which does not model exponential doubling.
- (b) P = 50 + 12n
- This equation represents linear growth with a slope of 12 and an intercept of 50, failing to account for the compounding nature of doubling.
- (c) P = 50 (2)^12n
- This equation implies the population doubles every $1/12$ of a year (or grows by a factor of $2^{12}$ per year), which contradicts the 12-year doubling period.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.