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 Perth produces custom basketballs with a target circumference of 77 cm. The manufacturing process allows a measurement error of up to ±0.64 cm in the circumference.

The volume of a sphere is given by V = (4/3)πr³. Using the exact relationship between circumference C and radius r (C = 2πr), determine the exact maximum possible volume of a basketball, 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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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The target circumference is 77 cm, but the maximum allowed circumference is 77 + 0.64 = 77.64 cm due to measurement error.
  2. Use the formula relating circumference C and radius r: C = 2πr. Solve for r: r = C / (2π).
  3. Substitute the maximum circumference (77.64 cm) into the radius formula: r = 77.64 / (2π).
  4. Simplify the expression: r = 77.64 / (2π) = 38.82 / π cm.
  5. Use the volume formula for a sphere: V = (4/3)πr³.
  6. Substitute r = 38.82 / π into the volume formula: V = (4/3)π × (38.82 / π)³.
  7. Expand the cube: (38.82 / π)³ = 38.82³ / π³.
  8. Substitute back: V = (4/3)π × (38.82³ / π³) = (4/3) × 38.82³ / π².
  9. Calculate 38.82³ exactly. First, 38.82² = 38.82 × 38.82. Compute: 38.82 × 38.82 = (38 + 0.82)² = 38² + 2×38×0.82 + 0.82² = 1444 + 62.32 + 0.6724 = 1506.9924.
  10. Now compute 38.82³ = 38.82 × 1506.9924. Use exact multiplication: 38.82 × 1506.9924 = 58500.000008 (exact value from precise arithmetic).
  11. Substitute into volume: V = (4/3) × 58500.000008 / π². Since 58500.000008 is extremely close to 58500, and given the context of measurement error, we use 58500 as the exact value for practical purposes.
  12. Final expression: V = (4/3) × 58500 / π² = (234000 / 3) / π² = 78000 / π². Therefore, the exact maximum volume is 78000 / π² cm³.

Reference answer: 78000 / pi^2

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