NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Irrational numbers

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A craft designer in Hobart is creating a square-based decorative panel for a public art installation. The area of the panel is 34 square metres.

Using the formula side = sqrt(area), what is the exact length of one side of the panel, in metres? Give your answer as a simplified exact value.

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Explanation

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Worked solution

  1. The area of the square panel is given as 34 square metres.
  2. The formula for the side length of a square given its area is side = sqrt(area).
  3. Substitute the given area into the formula: side = sqrt(34).
  4. Check if 34 is a perfect square. The perfect squares near 34 are 25 (5^2) and 36 (6^2). Since 34 is not a perfect square, sqrt(34) is irrational.
  5. Factor 34 to see if it has any square factors: 34 = 2 × 17. Neither 2 nor 17 is a perfect square, and there are no repeated prime factors.
  6. Since 34 has no square factors other than 1, sqrt(34) cannot be simplified further.
  7. Therefore, the exact length of one side is sqrt(34) metres.
  8. Verify by squaring the result: (sqrt(34))^2 = 34, which matches the given area.
  9. The value of sqrt(34) lies between 5.8 and 5.9 because 5.8^2 = 33.64 and 5.9^2 = 34.81.
  10. Since 34 is not a perfect square and has no square factors, the exact simplified form remains sqrt(34).
  11. The side length of the panel is exactly sqrt(34) metres.
  12. This matches the requirement for a simplified exact value.

Reference answer: sqrt(34)

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