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WMA12/01 Oct 2023, Question 6: The Trapezium Rule

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 6 marks

The question, in short

Estimate a river cross-section area from a depth curve using the trapezium rule, then find per-minute flow volume and judge if it is over or under the true value.

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

Part (a) · 3 marks

Apply the trapezium rule to the given depths at x = 0, 3, 6, 9, 12, 15 to estimate the cross-sectional area.

  1. 1Strip width h = 3 (x values 0, 3, 6, 9, 12, 15)
  2. 2y values: 0, 1.52, 2.74, 3.12, 3.08, 0
  3. 3Area ≈ (3/2) × {0 + 0 + 2(1.52 + 2.74 + 3.12 + 3.08)}
  4. 4= 1.5 × 20.92 = 31.38 m²

Answer31.38 m² (or 31.4 m²)

Part (b) · 2 marks

Combine the area with speed 1.5 m s⁻¹ to get the volume per minute in m³ to 2 significant figures.

  1. 1Volume per second = area × speed = 31.38 × 1.5
  2. 2Volume per minute = 31.38 × 1.5 × 60 = 2824.2
  3. 3≈ 2800 m³ (2 s.f.)

Answer2800 m³

Part (c) · 1 mark

Decide whether the part (b) volume overestimates or underestimates the real flow, justifying with the model's shape.

  1. 1The trapezium rule joins the measured points with straight lines
  2. 2The river bed curve lies beyond (deeper than) these straight lines
  3. 3So the trapezia cover less than the true cross-section: the area, and hence the volume, is an underestimate

AnswerUnderestimate, because the curve lies outside the tops of the trapezia (deeper than the straight lines), so the trapezia have a smaller area than the true cross-section

Stuck on a step? Read the free lesson: The Trapezium Rule

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Also searched as: WMA12 October 2023 · WMA12/01 · wma12 oct23 · IAS Maths Pure Mathematics 2 October 2023 · question 6.
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