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

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In a coastal engineering project in Sydney, a team measures the height of a sea wall using a clinometer and a tape measure. The tape measure records a horizontal distance of 15.75 metres from the base of the wall to the measurement point, with a maximum error of ±0.02 metres.

The clinometer records an angle of elevation of 58.3 degrees to the top of the wall, with a maximum error of ±0.05 degrees. The actual height h of the wall is calculated using the formula h = d × tan(θ), where d is the horizontal distance and θ is the angle of elevation.

The exact value of tan(58.3°) is known to be 1.6123 (for this context). What is the exact maximum possible value of h, in metres, when both d and θ are at their maximum measured values?

Express your answer as a simplified fraction.

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Explanation

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Worked solution

  1. The horizontal distance d is given as 15.75 metres with a maximum error of ±0.02 m. To find the maximum possible height, use the largest measured value of d: 15.75 + 0.02 = 15.77 m.
  2. The angle of elevation θ is given as 58.3 degrees with a maximum error of ±0.05°. To find the maximum possible height, use the largest measured value of θ: 58.3 + 0.05 = 58.35°.
  3. The height h is calculated using the formula h = d × tan(θ). The value of tan(58.35°) is not given directly, but we are told that tan(58.3°) = 1.6123. Since 58.35° is slightly larger than 58.3°, tan(58.35°) will be slightly larger than 1.6123. We must use the exact value of tan(58.35°) for accuracy.
  4. Using a calculator or trigonometric table, tan(58.35°) ≈ 1.6162 (accurate to 4 decimal places). This value is used in the calculation.
  5. Substitute d = 15.77 m and tan(θ) = 1.6162 into the formula: h = 15.77 × 1.6162.
  6. Perform the multiplication: 15.77 × 1.6162. First, compute 15.77 × 1.6162.
  7. Break down the multiplication: 15.77 × 1.6162 = (15 + 0.77) × 1.6162 = 15×1.6162 + 0.77×1.6162.
  8. Calculate 15 × 1.6162 = 24.243.
  9. Calculate 0.77 × 1.6162 = 1.244474.
  10. Add the two parts: 24.243 + 1.244474 = 25.487474 metres.
  11. The result must be expressed as a simplified fraction. Convert 25.487474 to a fraction. First, note that 25.487474 = 25487474 / 1000000. Simplify this fraction by dividing numerator and denominator by their greatest common divisor (GCD).
  12. After simplifying 25487474 / 1000000, the simplified fraction is 12743737 / 500000. This is the exact maximum possible value of h.

Reference answer: 12743737/500000

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