NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Modelling with linear relations

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A craftsperson in Brisbane is creating a decorative wooden frame in the shape of a parallelogram for a community art exhibition. The frame has a base length of 46 cm and a height of 52 cm.

The design requires that the perimeter of the frame be exactly 200 cm. The craftsperson uses a linear relation to model the side length (s) in terms of the base (b) and height (h), assuming opposite sides are equal and the frame is not a rectangle.

Using the linear model P = 2b + 2s, where P is the perimeter, b is the base, and s is the side length, what is the exact length of one of the non-base sides of the frame? Express your answer as an exact fraction.

Concept illustrations

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Translate a situation into a model. With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.$4 + $2/kmDefine tariff→C = 4 + 2dBuild rule→d=3 → C=$10Interpret
With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.
A straight-line relationship. Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.-2-2-1-111223344xy
Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The perimeter P of a parallelogram is given by the formula P = 2b + 2s, where b is the base length and s is the side length. This formula applies because opposite sides of a parallelogram are equal in length.
  2. The base length b is given as 46 cm. The height h is 52 cm, but height is not used in the perimeter formula because perimeter depends only on side lengths, not perpendicular height.
  3. The total perimeter P is required to be exactly 200 cm. Substitute P = 200 and b = 46 into the formula: 200 = 2(46) + 2s.
  4. Calculate 2 × 46: 2 × 46 = 92. The equation becomes 200 = 92 + 2s.
  5. Subtract 92 from both sides to isolate the term with s: 200 - 92 = 2s → 108 = 2s.
  6. Divide both sides by 2: 108 ÷ 2 = s → s = 54.
  7. The side length s is 54 cm. This is the length of one of the non-base sides, as the frame has two sides of length s and two of length b.
  8. The problem asks for the exact length as a fraction. Since 54 is an integer, it is equivalent to 54/1.
  9. Check the solution: 2 × 46 = 92, 2 × 54 = 108, 92 + 108 = 200. This matches the required perimeter.
  10. The frame is not a rectangle, so the side length must not equal the height. Since 54 ≠ 52, the condition is satisfied.
  11. The exact length of one of the non-base sides is 54 cm, expressed as the fraction 54/1.
  12. The answer is 54/1 cm, which simplifies to 54 cm, but as an exact fraction, it is 54/1.

Reference answer: 54/1 cm

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