Multiple choice

The population of a city increases at the rate $3\%$ per year. If at time t the population of city is p, then find equation of p in time t.

  1. $p=ce^{\displaystyle\frac{3t}{100}}$
  2. $p=3e^{\displaystyle\frac{3t}{100}}$
  3. $p=e^{\displaystyle\frac{3t}{100}}$
  4. $p=\displaystyle\frac{3}{100}e^{3t}$
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A Correct answer
Explanation

The rate of change of population is dp/dt = (3/100) * p. This is a separable differential equation: dp/p = (3/100) dt. Integrating both sides: ln(p) = (3t/100) + C. Exponentiating: p = e^(3t/100 + C) = c * e^(3t/100).

AI explanation

Using the differential equation for continuous exponential growth, dp/dt = 0.03p. Separating variables and integrating gives ln(p) = 0.03t + c, which is equivalent to ln(p) = (3t/100) + c. By taking the exponent of both sides, the equation becomes p = c * e^(3t/100).

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