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 Perth is modelling the spread rate of a controlled burn using direct proportion. The spread rate 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 10 km/h and the fuel load is 6 tonnes per hectare, the fire spreads at 240 m per hour. What is the exact spread rate, in metres per hour, when the wind speed is 15 km/h and the fuel load is 9 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 (in km/h) and fuel load (in tonnes per hectare). This means the relationship can be expressed as: spread rate = k × wind speed × fuel load, where k is the constant of proportionality.
  2. From the test scenario: wind speed = 10 km/h, fuel load = 6 tonnes per hectare, spread rate = 240 m per hour. Substitute these values into the formula to find k.
  3. 240 = k × 10 × 6
  4. 240 = k × 60
  5. Solve for k: k = 240 / 60 = 4
  6. Now use k = 4 to find the spread rate when wind speed = 15 km/h and fuel load = 9 tonnes per hectare.
  7. spread rate = 4 × 15 × 9
  8. 4 × 15 = 60
  9. 60 × 9 = 540
  10. The spread rate is 540 m per hour.
  11. Check: 15 km/h is 1.5 times 10 km/h, and 9 tonnes per hectare is 1.5 times 6 tonnes per hectare. So the spread rate should increase by 1.5 × 1.5 = 2.25 times. 240 × 2.25 = 540. Correct.
  12. The exact spread rate when wind speed is 15 km/h and fuel load is 9 tonnes per hectare is 540 m per hour.

Reference answer: 540

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