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4024/22 Oct/Nov 2022, Question 3: Reciprocal Functions

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 11 marks

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The question, in short

A cuboid with square base edge x and height h has volume 60 cm³; work with its surface area A, including a formula, table, graph, minimum, and a case where A = 120 cm².

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

Part (a) · 2 marks

Using the volume condition, derive the stated expression for the surface area A in terms of x.

  1. 1Volume: x²h = 60, so h = 60/x²
  2. 2A = 2x² + 4xh (two square ends, four rectangular sides)
  3. 3A = 2x² + 4x × 60/x² = 2x² + 240/x

Answerh = 60/x² (or xh = 60/x) seen, then A = 2x² + 4x × 60/x² → 2x² + 240/x or A = 2x² + 4 × 60/x → 2x² + 240/x

Part (b) · 2 marks

Calculate the missing A values for x = 5, 6, 7 in the table.

  1. 1x = 5: A = 2 × 25 + 240/5 = 50 + 48 = 98
  2. 2x = 6: A = 2 × 36 + 240/6 = 72 + 40 = 112

Answer| x | 5 | 6 | | A | 98 | 112 |

Part (c) · 3 marks

Plot the curve of A against x on the grid for x from 1 to 8.

  1. 1Plot (1, 242), (2, 128), (3, 98), (4, 92), (5, 98), (6, 112), (7, 132), (8, 158)
  2. 2Join with a smooth curve

AnswerCorrect smooth curve

Part (d) · 1 mark

Use your graph to state the smallest surface area the cuboid can have.

  1. 1Lowest point of the curve is near x = 3.9
  2. 2Minimum surface area ≈ 92 cm²

Answer90 to 92

Part (e) · 3 marks

Given A = 120 cm² and h larger than x, determine the cuboid's dimensions.

  1. 1Read the graph at A = 120 on the left branch (h > x means x is small): x ≈ 2.2
  2. 2h = 60/x² = 60/2.2² = 12.4
  3. 3Dimensions: 2.2 cm by 2.2 cm by 12.4 cm

Answerx = 2.1 to 2.3, h = 11.3 to 13.6

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