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 Darwin is using a proportional model to predict the spread rate of a controlled burn. The rate of fire spread is directly proportional to the wind speed and the fuel load.

In a test scenario, when the wind speed is 12 km/h and the fuel load is 8 tonnes per hectare, the fire spreads at 300 m per hour. What is the exact spread rate, in metres per hour, when the wind speed is 18 km/h and the fuel load is 10 tonnes per hectare?

Use direct proportion to determine the exact rate.

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Worked solution

  1. The fire spread rate is directly proportional to both wind speed and fuel load. This means the rate R is proportional to the product of wind speed (W) and fuel load (F), so R = k × W × F, where k is the constant of proportionality.
  2. From the test scenario, when W = 12 km/h and F = 8 t/ha, the spread rate R = 300 m/h. Substitute these values into the formula to find k: 300 = k × 12 × 8.
  3. Calculate the product of wind speed and fuel load: 12 × 8 = 96.
  4. Solve for k: k = 300 / 96. Perform division: 300 ÷ 96 = 3.125.
  5. Now use k = 3.125 to find the new spread rate when W = 18 km/h and F = 10 t/ha. Substitute into R = k × W × F: R = 3.125 × 18 × 10.
  6. First, multiply 18 × 10 = 180.
  7. Now multiply 3.125 × 180. Break it down: 3 × 180 = 540, and 0.125 × 180 = 22.5.
  8. Add the results: 540 + 22.5 = 562.5.
  9. The spread rate is 562.5 m/h. This is the exact value since 3.125 = 25/8 and 25/8 × 180 = (25 × 180) / 8 = 4500 / 8 = 562.5.
  10. Check: 300 / (12 × 8) = 300 / 96 = 3.125. Then 3.125 × 18 × 10 = 562.5. The calculation is consistent.
  11. The spread rate when wind speed is 18 km/h and fuel load is 10 t/ha is 562.5 metres per hour.
  12. Final answer confirmed: 562.5 m/h.

Reference answer: 562.5

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