O-LevelAdditional MathematicsLogarithmic and exponential functions4037/21 · Oct/Nov 2010 · Q35 Marks
A number, N₀, of fish of a particular species are introduced to a lake. The number, N, of these fish in the lake, t weeks after their introduction, is given by N = N₀e⁻ᵏᵗ, where k is a constant. Calculate
- (i)the value of k if, after 34 weeks, the number of these fish has fallen to ¹/₂ of the number introduced,[2]
- (ii)the number of weeks it takes for the number of these fish to have fallen to ¹/₅ of the number introduced.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2010). The mark or mark type is shown in brackets.
- [M1] (i) Solve 0.5 = e⁻³⁴ᵏ using ln or log correctly
- [A1] (i) k = 0.0204 or ¹/₃₄ ln 2
- [B1] (ii) e⁻ᵏᵗ = 5 or e⁻ᵏᵗ = 0.2 with k numerical
- [M1] (ii) t = ¹/ᵏ ln 5 with k numerical
- [A1] (ii) 79
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