A-LevelMathematicsLogarithmic and exponential functions9709/33 · Oct/Nov 2024 · Q35 Marks

The number of bacteria in a population, P, at time t hours is modelled by the equation P = aekt, where a and k are constants. The graph of InP against t, shown in the diagram, has gradient 1/20 and intersects the vertical axis at (0,3).

(a)State the value of k and find the value of a correct to 2 significant figures.[3]
(b)Find the time taken for P to double. Give your answer correct to the nearest hour.[2]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[B1] (a) State or imply that ln P = ln a + kt or ln P = ln a + k(ln e)t.
[B1] (a) State k = 1/20.
[B1] (a) ln a = 3 ⇒ a = 20 to 2 sf.
[M1] (b) Form a correct equation in t using a and k, or their a and k where a will cancel (or are both numerical). E.g. 2a = aekt, 2=ekt, kt = ln 2.
[A1] (b) Obtain t = 14 [hours].

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