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0580/42 Oct/Nov 2022, Question 2: Drawing Straight-Line Graphs

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 12 marks

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The question, in short

For a cuboid with square base side x and height 9 − x, work with volume V = x²(9 − x), complete a table, plot its graph, solve for x at a given volume, and handle surface area.

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

Part (ai) · 1 mark

Fill in the missing V value in the table for x = 2.

  1. 1V = 2² × (9 − 2) = 4 × 7 = 28

Answer28

Part (aii) · 4 marks

Plot the curve of V against x from 0 to 9 on the grid.

  1. 1Plot (0, 0), (1, 8), (2, 28), (3, 54), (4, 80), (5, 100), (6, 108), (7, 98), (8, 64), (9, 0)
  2. 2Join with a smooth curve, maximum near (6, 108)

AnswerCorrect curve

Part (aiii) · 2 marks

Determine the x values giving a cuboid volume of 44 cm³.

  1. 1Draw the line V = 44 and read off where it meets the curve
  2. 2x²(9 − x) = 44 gives x ≈ 2.63 and x ≈ 8.37

Answer2.5 to 2.8, 8.2 to 8.5

Part (bi) · 2 marks

Prove that the cuboid's total surface area equals (36x − 2x²) cm².

  1. 1Two square faces: 2 × x² = 2x²
  2. 2Four rectangular faces: 4 × x(9 − x) = 36x − 4x²
  3. 3Total = 2x² + 36x − 4x² = 36x − 2x²

Answer2x² + 4x(9 − x) = 2x² + 36x − 4x², leading to 36x − 2x²

Part (bii) · 3 marks

Using maximum volume 108 cm³ at x = 6, calculate the surface area at that maximum.

  1. 1Maximum volume at x = 6 (from the graph)
  2. 2Surface area = 36 × 6 − 2 × 6² = 216 − 72
  3. 3= 144 cm²

Answer144

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Also searched as: 0580/42/O/N/22 · 0580_w22_qp_42 · 0580 w22 42 · question 2.
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