NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Accuracy and measurement errors

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A craftsperson in Melbourne is designing a triangular wooden inlay for a decorative panel. The inlay is to be cut from a piece of timber with two sides measuring exactly 67 cm and 89 cm, and the included angle between them is exactly 60 degrees.

Using the formula for the area of a triangle given two sides and the included angle, A = (1/2) * a * b * sin(C), determine the exact area of the triangular inlay in square centimetres. Express your answer as an exact simplified fraction or radical expression.

Concept illustrations

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Measurement compared with a reference. Measured length 49 cm differs from the reference 50 cm by 1 cm. Absolute error is 1 cm and percentage error is 2%. The axis is a close-up from 48 to 51 cm.48 cm49 cm50 cm51 cmMeasured: 49Reference: 50|49 − 50| = 1 cm
Measured length 49 cm differs from the reference 50 cm by 1 cm. Absolute error is 1 cm and percentage error is 2%. The axis is a close-up from 48 to 51 cm.
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.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The triangle has two sides of 67 cm and 89 cm, with an included angle of 60 degrees. The area of a triangle given two sides and the included angle is calculated using the formula A = (1/2) * a * b * sin(C).
  2. Substitute the given values into the formula: a = 67 cm, b = 89 cm, C = 60°.
  3. The sine of 60 degrees is a known exact value: sin(60°) = sqrt(3)/2.
  4. Substitute sin(60°) into the formula: A = (1/2) * 67 * 89 * (sqrt(3)/2).
  5. Multiply the constants: (1/2) * 67 * 89 * (sqrt(3)/2) = (67 * 89 * sqrt(3)) / 4.
  6. Calculate 67 * 89 using multiplication: 67 * 89 = 67 * (90 - 1) = 67*90 - 67 = 6030 - 67 = 5963.
  7. Substitute the product: A = 5963 * sqrt(3) / 4.
  8. The expression 5963 * sqrt(3) / 4 is already in simplified radical form. No further simplification is possible because 5963 and 4 share no common factors (5963 is odd, so not divisible by 2).
  9. The area is therefore (5963 * sqrt(3)) / 4 square centimetres.
  10. Check the calculation: 67 * 89 = 5963 (verified via 60*89 = 5340, 7*89 = 623, 5340 + 623 = 5963).
  11. The formula correctly applies because two sides and the included angle are known, and the sine of 60° is exact.
  12. The exact area of the triangular inlay is (5963 * sqrt(3)) / 4 cm².

Reference answer: 5963 * sqrt(3) / 4

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