NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Proportion, rates and scale modelling

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A fire management team in Canberra is modelling the spread rate of a controlled burn using direct proportion. The spread rate (in metres per hour) is directly proportional to both wind speed (in km/h) and fuel load (in tonnes per hectare).

In a test scenario, when the wind speed is 14 km/h and the fuel load is 7 tonnes per hectare, the fire spreads at 350 m per hour. What is the exact spread rate, in metres per hour, when the wind speed is 21 km/h and the fuel load is 12 tonnes per hectare?

Use the proportional relationship to calculate the exact rate.

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Worked solution

  1. The spread rate is directly proportional to both wind speed and fuel load, so the relationship is: spread rate = k × wind speed × fuel load, where k is the constant of proportionality.
  2. Given: wind speed = 14 km/h, fuel load = 7 t/ha, spread rate = 350 m/h.
  3. Substitute known values into the formula to find k: 350 = k × 14 × 7.
  4. Calculate 14 × 7 = 98.
  5. Now solve for k: k = 350 / 98.
  6. Simplify 350 / 98: divide numerator and denominator by 14 to get 25 / 7.
  7. Now k = 25 / 7 (in units of m/h per (km/h × t/ha)).
  8. New scenario: wind speed = 21 km/h, fuel load = 12 t/ha.
  9. Use the formula: spread rate = k × 21 × 12.
  10. Substitute k = 25 / 7: spread rate = (25 / 7) × 21 × 12.
  11. First, 21 / 7 = 3, so (25 / 7) × 21 = 25 × 3 = 75.
  12. Then 75 × 12 = 900. So the spread rate is 900 m/h.

Reference answer: 900 m/h

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