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4PM1/02R Jan 2020, Question 6: Connected Rates of Change and Small Changes

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 6 marks

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The question, in short

Cone with vertical angle 60° drains at 0.9 cm³/s; liquid height h and volume V. Show the volume formula and find how fast h falls when h is 1.2 cm.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

Using the 60° vertical angle, relate the surface radius r to h, then show the cone volume reduces to V = (1/9)πh³.

  1. 1Half the vertical angle is 30°, so r = h tan 30° = h/√3
  2. 2V = ⅓πr²h = ⅓π(h²/3)h
  3. 3V = (1/9)πh³

AnswerV = (1/9)πh³ shown

Part (b) · 4 marks

Differentiate V with respect to t, use dV/dt = −0.9, and find dh/dt at h = 1.2 cm to 3 significant figures.

  1. 1dV/dh = (1/3)πh²
  2. 2dh/dt = dh/dV × dV/dt = (3/(πh²)) × 0.9
  3. 3h = 1.2: dh/dt = 2.7/(π × 1.44) = 0.5968…
  4. 4Height decreasing at 0.597 cm/s

Answer0.597 cm/s

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