NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Exact and approximate calculations

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A craft designer in Brisbane is creating a decorative panel consisting of a square with side length 39 cm and an equilateral triangle with side length 59 cm attached to one side of the square, such that the base of the triangle coincides with a side of the square. The area of an equilateral triangle with side length s is given by the formula A = (sqrt(3)/4) * s^2.

Calculate the exact total area of the compound shape in square centimetres, expressed as a simplified exact expression involving sqrt(3).

Concept illustrations

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Nearest hundredth on a number line. 6.347 lies between 6.34 and 6.35. It is beyond the midpoint 6.345, so rounding to the nearest hundredth gives 6.35.6.346.3456.356.3476.347 ≈ 6.35
6.347 lies between 6.34 and 6.35. It is beyond the midpoint 6.345, so rounding to the nearest hundredth gives 6.35.
Rational numbers inside the real numbers. Rational and irrational numbers are disjoint parts of the real numbers. The examples 4 and 1/2 are rational; √2 and π are irrational.Real numbersRationalIrrational4 1/2√2 π
Rational and irrational numbers are disjoint parts of the real numbers. The examples 4 and 1/2 are rational; √2 and π are irrational.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The compound shape consists of a square with side length 39 cm and an equilateral triangle with side length 59 cm attached along one side of the square.
  2. The area of a square is given by A = s^2, where s is the side length. Substitute s = 39 cm.
  3. Calculate the area of the square: 39^2 = 39 × 39. 39 × 39 = (40 - 1)^2 = 1600 - 80 + 1 = 1521 cm².
  4. The area of an equilateral triangle is given by A = (sqrt(3)/4) × s^2, where s is the side length. Substitute s = 59 cm.
  5. Calculate s^2: 59^2 = (60 - 1)^2 = 3600 - 120 + 1 = 3481 cm².
  6. Substitute into the formula: A = (sqrt(3)/4) × 3481 = (3481 × sqrt(3)) / 4 cm².
  7. The total area is the sum of the square and triangle areas: 1521 + (3481 × sqrt(3)) / 4 cm².
  8. Express the total area as a single simplified exact expression: 1521 + (3481/4) × sqrt(3) cm².
  9. Check if 3481 and 4 have common factors. 3481 is odd, so not divisible by 2. 3481 ÷ 4 = 870.25, not an integer. No simplification possible.
  10. The expression 1521 + (3481/4) × sqrt(3) is already in simplified exact form, as 3481 and 4 are coprime and sqrt(3) is irrational.
  11. Verify the area of the square: 39 × 39 = 1521 cm² (correct).
  12. The total area of the compound shape is 1521 + (3481/4) × sqrt(3) cm².

Reference answer: 1521 + (3481/4)*sqrt(3) cm²

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