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  • Ahm97sic's Avatar
    346 posts since Apr '08
    • For "O" level Add maths students to practise

      Question 5(a) (from Pan Pacific Additional Mathematics Page 421)

      In an experiment, the growth rate of bacteria in a liquid at any time is proportional to N, the number of bacteria after t seconds. Given that the initial number of bacteria is A, show that N = Ae^(kt) , where k is a constant.

      Steps for Solution for Q5(a)

      (1)    Given that the initial number of bacteria is A, show that N = Ae^(kt) ,

               where k is a constant.

      (2)     Since the growth rate of bacteria (ie dN/dt) in a liquid at any time

               is proportional to N (ie kN), the number of bacteria after t seconds.

               So, dN/dt = kN

      (3)    dN/dt = kN

              dN/N = kdt

              (1/N) dN = k dt

              Integrate on both sides

              Integrate (1/N) dN = Integrate k dt

              ln N + C1 = kt + C2

              ln N = kt + C2 - C1

                  N = e ^ ( kt + C2 - C1)

                 N = e ^ ( C2 - C1 ) e ^ (kt)

      (4)   Initial number of bacteria is A ie when t = 0 , N = A

              Substitute t = 0 , N = A into N = e ^ ( C2 - C1 )  e ^ (kt)

                                                       A = e ^ ( C2 - C1 ) e^ (k x 0)

                                                       A = e ^ ( C2 - C1 ) e^0

                                                       A = e ^ ( C2 - C1 ) x 1    since e^0 = 1

                                                       A = e ^ ( C2 - C1 )

      (5)    Substitute A = e ^ ( C2 - C1 ) into N = e ^ ( C2 - C1 ) e ^ (kt)

                                                                N = Ae^(kt)  (Shown)

      Thank you for your kind attention.

      Regards,

      ahm97sic

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