A cycle-hire scheme used by public transport commuters charges a fixed cost plus an hourly rate. The table below shows the cost, \(C\) pounds, of hiring a b...
Question 1 Report
A cycle-hire scheme used by public transport commuters charges a fixed cost plus an hourly rate. The table below shows the cost, \(C\) pounds, of hiring a bike for \(t\) whole hours.
| t (hours) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| C (£) | 3.20 | 5.40 | 7.60 | 9.80 |
- Find the values of \(a\) and \(b\) in the formula \(C = at + b\). (2)
- Work out the cost of hiring a bike for \(9\) hours. (1)
Answer Details
(a) The cost rises by the same amount, \(\pounds2.20\), for every extra hour (\(5.40-3.20=2.20\), \(7.60-5.40=2.20\), and \(9.80-7.60=2.20\)), so the gradient of the formula \(C=at+b\) is \(a=2.2\). Substituting the point \(t=1\), \(C=3.20\): \(2.2(1)+b=3.2\), so \(b=1\). [2 marks]
(b) With \(C=2.2t+1\), substituting \(t=9\): \(C=2.2(9)+1=19.8+1=\pounds20.80\). [1 mark]
The constant \(b=1\) is what the formula would predict as a fixed base cost even before any hourly charge is added, while \(a=2.2\) is the cost of each additional hour beyond that; extending the pattern in the table by \(5\) more hours from \(t=4\) (\(\pounds9.80+5\times\pounds2.20=\pounds20.80\)) gives the same answer as a useful check.
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