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A construction company is designing a staircase that complies with the Australian Building Code, which specifies that the sum of the rise (vertical height) and tread (horizontal depth) of each step must not exceed 550 mm, and the rise must be between 150 mm and 200 mm. A designer proposes a staircase with a rise of 180 mm per step.
Using direct proportion, determine the exact tread depth in millimetres that satisfies the building code constraint. Express your answer as an exact fraction or integer.
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Worked solution
- The Australian Building Code requires that the sum of the rise and tread of each step must not exceed 550 mm.
- The rise per step is given as 180 mm, which is within the allowed range of 150 mm to 200 mm.
- Let the tread depth be t mm. According to the code, rise + tread ≤ 550 mm.
- Substitute the known rise: 180 + t ≤ 550.
- Solve for t: t ≤ 550 - 180.
- Calculate: 550 - 180 = 370.
- Therefore, t ≤ 370 mm.
- The problem asks for the exact tread depth that satisfies the constraint using direct proportion. Since the sum must not exceed 550 mm and the rise is fixed at 180 mm, the maximum allowed tread is 370 mm.
- There is no indication that the sum must be exactly 550 mm, but the question specifies 'determine the exact tread depth' that satisfies the constraint. Since the rise is fixed and the sum must not exceed 550 mm, the maximum valid tread is 370 mm.
- The problem states to use direct proportion. However, since the rise is fixed and the sum is capped, the tread is determined by subtraction, not proportionality. But the only value that satisfies the sum constraint exactly and maximises the tread (as implied by 'exact' in context) is 370 mm.
- Check: 180 mm rise + 370 mm tread = 550 mm, which is exactly the maximum allowed sum.
- Thus, the exact tread depth that satisfies the building code constraint is 370 mm.
Reference answer: 370
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