Multiple choice

If the rate of increase of the population of the city is $5\%$ per year. In t time population of city is P, then expression of P in terms of t is _________.

  1. $P=3\cdot e^{t/20}$
  2. $P=e^{t/20}$
  3. $P=c\cdot e^{t/20}$
  4. $P=\dfrac{1}{20}e^{5t}$
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C Correct answer
Explanation

Population growth at a continuous rate r is given by P = P0 * e^(rt). Here, r = 5% = 0.05 = 1/20. Thus, P = c * e^(t/20), where c is the initial population.

AI explanation

Using the formula for exponential population growth, dP/dt = 0.05P. Integrating this differential equation gives ln(P) = 0.05t + c, which can be rewritten as ln(P) = t/20 + c. Taking the exponential of both sides yields the expression P = c * e^(t/20).

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