Ex 9.4, 22 - In a culture, bacteria count is 1,00,000. Number - Variable separation - Statement given

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  1. Chapter 9 Class 12 Differential Equations
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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, 𝑑𝑦﷮𝑑𝑡﷯ ∝ y 𝑑𝑦﷮𝑑𝑡﷯ = ky 𝑑𝑦﷮𝑦﷯ = kdt Integrating both sides ﷮﷮ 𝑑𝑦﷮𝑦﷯=𝑘﷯ ﷮﷮𝑑𝑡﷯ 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 = 𝟐 𝐥𝐨𝐠﷮𝟐﷯﷮ 𝐥𝐨𝐠﷮ 𝟏𝟏﷮𝟏𝟎﷯﷯﷯﷯

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