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WMA12/01 Jan 2025, Question 3: Stationary Points, Increasing and Optimising

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 10 marks

The question, in short

Open cuboid container has base 3x by x, height y, capacity 120 m³. Find constants in the surface area formula, then use calculus to locate and verify the minimum.

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

Part (a) · 4 marks

Use the 120 m³ capacity to express y in terms of x, then show the sheet-metal area A equals Px² + Q/x and state P and Q.

  1. 1Volume: 3x² y = 120, so y = 40/x²
  2. 2Area (open top): A = 3x² + 2(3xy) + 2(xy) = 3x² + 8xy
  3. 3Substitute y: A = 3x² + 8x × 40/x²
  4. 4A = 3x² + 320/x, so P = 3, Q = 320

AnswerShown: A = 3x² + 320/x, so P = 3 and Q = 320

Part (b) · 4 marks

Differentiate A with respect to x, set the derivative to zero, and solve for the stationary x value to 3 significant figures.

  1. 1dA/dx = 6x − 320/x²
  2. 2Set dA/dx = 0: 6x = 320/x²
  3. 3x³ = 160/3
  4. 4x = 3.76 (3 s.f.)

Answerx = 3.76

Part (c) · 2 marks

Find the second derivative of A, evaluate it at the x from part (b), and show it is positive, confirming a minimum.

  1. 1d²A/dx² = 6 + 640/x³
  2. 2At x = 3.76: d²A/dx² = 6 + 640/3.76³ ≈ 18
  3. 318 > 0, so A has a minimum at x = 3.76

AnswerShown: d²A/dx² = 6 + 640/x³ ≈ 18 > 0 at x = 3.76, so A is a minimum

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Also searched as: WMA12 January 2025 · WMA12/01 · wma12 jan25 · IAS Maths Pure Mathematics 2 January 2025 · question 3.
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