A bike shop calculates the recommended tyre pressure, \(p\) psi, for a given tyre size as directly proportional to the rider's weight, \(w\) kg. A rider wei...
Question 1 Report
A bike shop calculates the recommended tyre pressure, \(p\) psi, for a given tyre size as directly proportional to the rider's weight, \(w\) kg. A rider weighing 70 kg needs a pressure of 63 psi.
- Find the constant of proportionality. (2)
- Find the recommended pressure for a rider weighing 90 kg. (2)
- The manufacturer states the pressure must not exceed 85 psi. State whether the 90 kg rider's tyre is safe, with a reason. (1)
Answer Details
Direct proportion between pressure and weight means pressure equals a constant, \(k\), multiplied by weight; \(k\) is found from the known rider before being used to predict the pressure for a different rider and check it against the safety limit.
- \[ p = kw \], and substituting \(w=70\), \(p=63\): \[ k = 63\div70 = 0.9 \] [2 marks]
- With \(w=90\): \[ p = 0.9\times90 = 81 \text{ psi} \] [2 marks]
- Since \(81 \lt 85\), the tyre pressure is safe for the \(90\) kg rider. [1 mark]
Because pressure is directly, not inversely, proportional to weight here, a heavier rider needs a higher recommended pressure, not a lower one, which is why the pressure rises from \(63\) psi to \(81\) psi as the weight increases from \(70\) kg to \(90\) kg.
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