The rate of increase of the population of a city is proportional to the population present at that instant. In the period of 40 years the population increased from 30,000 to 40,000 . At any time t the population is (a)(b)^ t 40 . Then the value of a and b are respectively

Step-by-step solution

The population growth follows the differential equation dP dt = kP , leading to the solution P(t) = P₀ e^ kt with P₀ = 30,000 . Using the condition P(40) = 40,000 , we get 40,000 = 30,000 e^ 40k , so e^ 40k = 4 3 . Rewriting P(t) = 30,000 e^ kt as P(t) = 30,000 (e^ 40k )^ t 40 , then P(t) = 30,000 ( 4 3 )^ t 40 . This matches the form (a)(b)^ t 40 with a = 30,000 and b = 4 3 , corresponding to option B. Final answer: B