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

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A sports equipment manufacturer in Brisbane produces custom rugby balls with a target circumference of 81 cm. The measurement process has a maximum error of ±13 cm in the circumference.

The volume of a sphere is given by V = (4/3)πr³, and the relationship between circumference C and radius r is C = 2πr. What is the exact maximum possible volume of a rugby ball, in cubic centimetres, when the measured circumference is at its maximum value?

Express your answer as an exact simplified fraction involving π.

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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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The target circumference is 81 cm, but measurement error allows a maximum deviation of ±13 cm. The maximum measured circumference is 81 + 13 = 94 cm.
  2. The circumference C is related to the radius r by C = 2πr. To find r, rearrange: r = C / (2π).
  3. Substitute the maximum circumference (94 cm) into the formula: r = 94 / (2π).
  4. Simplify: r = 94 / (2π) = 47 / π cm.
  5. The volume of a sphere is V = (4/3)πr³. Substitute r = 47 / π into the formula.
  6. V = (4/3)π × (47 / π)³.
  7. Expand (47 / π)³: (47³) / (π³) = 103823 / π³.
  8. Now substitute: V = (4/3)π × (103823 / π³).
  9. Simplify by cancelling one π: V = (4/3) × (103823 / π²).
  10. Multiply: (4 × 103823) / (3 × π²) = 415292 / (3π²).
  11. The expression 415292 / (3π²) is the exact maximum volume in cubic centimetres. It is already simplified as a fraction involving π.
  12. The maximum possible volume is exactly 415292 / (3π²) cm³.

Reference answer: 415292 / (3π²) cm³

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