NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Irrational numbers

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A sports equipment manufacturer in Perth is producing square-shaped training mats for a new gym. Each mat has an area of 33 square metres.

What is the exact length of one side of the mat, in metres? Use the formula: side = sqrt(area).

Give your answer as a simplified exact value.

Concept illustrations

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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.
A square with area forty-nine. 49 unit squares form a 7 by 7 square. √49 = 7.7 units across7
49 unit squares form a 7 by 7 square. √49 = 7.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The area of the square training mat is given as 33 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(33).
  4. Check if 33 can be simplified under the square root. Factor 33: 33 = 3 × 11.
  5. Neither 3 nor 11 is a perfect square, and there are no repeated prime factors, so sqrt(33) cannot be simplified further.
  6. Therefore, the exact side length is sqrt(33) metres.
  7. Verify that sqrt(33) is irrational: 33 is not a perfect square, and its square root cannot be expressed as a fraction of integers.
  8. Confirm that squaring sqrt(33) gives 33: (sqrt(33))^2 = 33, which matches the given area.
  9. The value of sqrt(33) lies between 5 and 6, since 5^2 = 25 and 6^2 = 36, and 33 is between them.
  10. No rational number squared gives 33, so the exact value must be irrational and expressed as sqrt(33).
  11. The exact length of one side of the mat is sqrt(33) metres.
  12. This matches the requirement for a simplified exact value.

Reference answer: sqrt(33)

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