At a certain reference year, the world population growth rate was 3.5%.…
2013
At a certain reference year, the world population growth rate was 3.5%. Assuming exponential population growth, after approximately how many years would the world population increase by a factor of 16?
Answer: A. ~ 80 years — CONCEPTFor exponential growth at a constant continuous rate r, the population follows P(t) = P0ert. For a multiplication factor F, the elapsed time is t =…
- A.
~ 80 years
- B.
~ 40 years
- C.
~ 160 years
- D.
~ 320 years
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Correct answer: A
CONCEPT
For exponential growth at a constant continuous rate r, the population follows P(t) = P0ert.
For a multiplication factor F, the elapsed time is t = ln(F)/r. A factor of 16 is also four doublings because 16 = 24.
APPLICATION
Convert the growth rate: r = 3.5% = 0.035 per year.
Set the required factor: P(t)/P0 = 16 = e0.035t.
Take natural logarithms: 0.035t = ln(16) = 4 ln(2) ≈ 2.7726.
Solve for time: t = 2.7726/0.035 ≈ 79.2 years, which rounds to approximately 80 years.
CROSS-CHECK
The doubling time is ln(2)/0.035 ≈ 19.8 years. Four doublings take about 4 × 19.8 = 79.2 years, confirming the calculation.
RESULT
Therefore, the population becomes about 16 times as large after approximately 80 years.