2025 BVSS PRELIMS P2 Q1

2025 BVSS PRELIMS P2 Q1

Secondary 4
8 marks
2025 Bedok View Secondary School Prelims A Math Paper 2

The nutrient absorption rate, \(N\), in mg/hour, of a plant depends on the diameter \(x\), in mm, of its root hairs and is modelled by the function \(N(x)=x^2\ln\left(\frac{2}{x}\right)\), where \(0<x<2\).

  1. Find \(N'(x)\).[3]
  2. Find the optimal root hair diameter that maximises nutrient absorption and show that the nutrient absorption is a maximum.[5]

Solution:

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Answer:(a) \(N'(x)=2x\ln\left(\frac{2}{x}\right)-x\) (b) Optimal diameter \(=\frac{2}{\sqrt e}=1.21\text{ mm}\). Since \(N''\left(\frac{2}{\sqrt e}\right)=-2<0\), the nutrient absorption is a maximum. \(N_{\max}=\frac{2}{e}=0.736\text{ mg/hour}\).

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