Multiple choice

In planning maintenance for a city's infrastructure, a civil engineer estimates that, starting from the present, the population of the city will decrease by $10$ percent every $20$ years. If the present population of the city is $50,000$, which of the following expressions represents the engineers estimate of the population of the city $t$ years from now?

  1. $50,000(0.1)^{2t}$
  2. $50,000(0.1)^{\tfrac {t}{2}}$
  3. $50,000(0.9)^{2t}$
  4. $50,000(0.9)^{\tfrac {t}{20}}$
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D Correct answer
Explanation

The population decreases by 10% every 20 years, meaning it is multiplied by 0.9 every 20 years. For t years, the multiplier is (0.9)^(t/20).

AI explanation

The engineer uses the exponential decay formula A = P(1 - r)^(t/n), where P is the initial value, r is the rate, t is the total time, and n is the interval length. Since the population decreases by 10%, the decay factor is 0.9, and it happens every 20 years. The number of decay periods for t years is the fraction t/20. Substituting these values gives the expression 50,000(0.9)^(t/20).

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