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4MA1/1H Nov 2021, Question 19: Composite Perimeter, Area and Volume

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 6 marks

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The question, in short

A five-sided shape made from rectangle ABCD and equilateral triangle CED has perimeter 100 cm; derive the area formula and find the x giving maximum area.

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

Part (a) · 3 marks

Using AB = x and BC = y, express the area R of the shape and show it equals (x/4)(200 − [6 − √3]x).

  1. 1Perimeter: x + y + y + x + x = 3x + 2y = 100, so y = (100 − 3x)/2
  2. 2Rectangle area = xy = x(100 − 3x)/2
  3. 3Equilateral triangle area = 1/2 × x × x × sin 60° = (√3/4)x²
  4. 4R = x(100 − 3x)/2 + (√3/4)x² = (200x − 6x² + √3x²)/4
  5. 5R = (x/4)(200 − [6 − √3]x)

AnswerR = x(200 − 6x + x√3)/4

Part (bi) · 2 marks

Determine the x value maximising R, giving it as p/(q − √3) with integers p and q.

  1. 1R = (200x − (6 − √3)x²)/4
  2. 2dR/dx = (200 − 2(6 − √3)x)/4 = 0
  3. 3200 = 2(6 − √3)x
  4. 4x = 100/(6 − √3)

Answer100/(6 − √3)

Part (bii) · 1 mark

Justify why this x value produces the maximum area R.

  1. 1R = (200x − (6 − √3)x²)/4 is a quadratic in x
  2. 26 − √3 is positive, so the x² coefficient is negative
  3. 3The graph is an upside-down U, so the turning point is a maximum

AnswerThe graph of R is a quadratic in x with a negative coefficient of x², so it has a maximum point.

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