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4PM1/01 Jun 2019, Question 6: Tangents, Normals and Stationary Points (Further)

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 11 marks

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The question, in short

Lawn made of rectangle ABDE with length y and width 2x, plus semicircles of radius x at each end, perimeter 90 m and area S. Express S, then maximise it.

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Part (a) · 4 marks

Show S can be written as kx − πx² and state the constant k.

  1. 1Perimeter: 2y + 2πx = 90
  2. 2Area: S = 2xy + πx²
  3. 32y = 90 − 2πx, so S = x(90 − 2πx) + πx²
  4. 4S = 90x − πx², k = 90

AnswerS = 90x − πx² shown, k = 90

Part (b) · 5 marks

Use calculus to find x, to 4 significant figures, giving maximum S, and justify it is a maximum.

  1. 1dS/dx = 90 − 2πx
  2. 2dS/dx = 0 → x = 90/(2π) = 14.32 (4 s.f.)
  3. 3d²S/dx² = −2π < 0, so S is a maximum

Answerx = 14.32; d²S/dx² = −2π < 0 so maximum

Part (c) · 2 marks

Find the maximum value of S to the nearest whole number.

  1. 1S = 90(90/(2π)) − π(90/(2π))²
  2. 2= 644.57…
  3. 3= 645 m² (nearest whole number)

Answer645 m² (or 644)

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