Ex 9.4, 22 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.4, 22 In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours, In how many hours will the count reach 2,00,000 , if the rate of growth of bacteria is proportional to the number present? Let the number of bacteria at time t be y Given that, y = ky = kdt Integrating both sides = log y = kt + C According to Questions Now, Putting t = 0 and y = 1,00,000 in (1) log 1,00,000 = k 0 + C C = log 1,00,000 Putting value of C in (1) log y = kt + log 1,00,000 Now, Putting t = 2 and y = 1,00,000 in (2) log 1,10,000 = 2k + log 1,00,000 log 1,10,000 log 1,00,000 = 2k log 1,10,000 1,00,000 = 2k 1 2 log 11 10 = k Putting value of k in (2) log y = 1 2 log 11 10 t + log 1,00,000 Now, If Bacterial = 2,00,000, we have to find t Putting y = 2,00,000 in (3) log 2,00,000 = 1 2 log 11 10 t + log (1,00,000) log 2,00,000 log 1,00,000 = 1 2 log 11 10 t log 2,00,000 1,00,000 = 1 2 log 11 10 t log 2 = 1 2 log 11 10 t t =