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4PM1/01 Jun 2020, Question 10: Connected Rates of Change and Small Changes

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 12 marks

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The question, in short

Prove the curved surface area formula for a cone using a sector, then analyse a growing sand pile cone with height three times its radius to find a constant and a rate of change.

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

Part (a) · 4 marks

By folding a sector with radius R and arc length L into the cone's curved surface, show that A = πrl.

  1. 1The arc length L becomes the base circumference: L = 2πr
  2. 2R = l, and L = lθ ⇒ θ = 2πr/l
  3. 3A = ½l²θ = ½l² × 2πr/l
  4. 4= πrl

AnswerA = πrl proved

Part (b) · 3 marks

Given dA/dr = kπr and height equals three times radius, determine the exact constant k.

  1. 1h = 3r ⇒ l = √(r² + 9r²) = r√10
  2. 2A = πr × r√10 = √10 πr²
  3. 3dA/dr = 2√10 πr ⇒ k = 2√10

Answerk = 2√10

Part (c) · 5 marks

Find the rate at which curved surface area grows, in cm²/s to 3 s.f., when height is 24 cm.

  1. 1dV/dt = 1.5
  2. 2V = (1/3)πr² × 3r = πr³ ⇒ dV/dr = 3πr²
  3. 3h = 24 ⇒ r = 8
  4. 4dA/dt = dA/dr × dr/dV × dV/dt = 2√10 π × 8 × 1/(3π × 64) × 1.5
  5. 5= 0.3952… ≈ 0.395 cm²/s

Answer0.395 cm²/s

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