2024 CJC H1 Q5
2024 CJC H1 Q5
14 marks
A warehouse has 900 square metres of storage space.
- The total storage space needed for one consignment of children bicycles, one consignment of mountain bicycles and one consignment of road bicycles is \(635\) square metres.
- The storage space for one consignment of road bicycles is \(35\) square metres less than the total storage space for one consignment of children bicycles and one consignment of mountain bicycles.
- Two consignments of children bicycles, one consignment of mountain bicycles and two consignments of road bicycles exceed the warehouse storage space by \(265\) square metres.
- Write this information as three linear equations, and hence find the storage space needed for each consignment of children bicycles, mountain bicycles and road bicycles.[4]
- If the warehouse receives three consignments of road bicycles and one consignment of children bicycles, find the total storage space required. By considering the total storage space needed, what kind of advice would you give the warehouse manager?[2]
A company produces road bicycles. The manufacturing cost of one road bicycle, \(R\) dollars, can be modelled by
\[R = \frac{5000 + 23x}{x}\]
where \(x\) is the number of road bicycles produced in a month.
- Find the value of \(R\) when \(x = 100\) and explain in context what this value represents.[2]
- Sketch the graph of \(R\) against \(x\).[2]
- Find \(\frac{\mathrm{d}R}{\mathrm{d}x}\) and explain in context what the expression for \(\frac{\mathrm{d}R}{\mathrm{d}x}\) indicates about the manufacturing cost in the future.[2]
- The company sells each road bicycle for \(75\). By adding a suitable graph to your sketch in part (d), find the least number of road bicycles the company needs to sell in a month to make a profit.[2]
Solution:
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Answer:(a) Children: \(230\ \mathrm{m}^2\), mountain: \(105\ \mathrm{m}^2\), road: \(300\ \mathrm{m}^2\) per consignment.
(b) \(1130\ \mathrm{m}^2\); arrange \(230\ \mathrm{m}^2\) extra storage or stagger deliveries.
(c) 73 dollars per bicycle when 100 are produced per month.
(d) Sketch shown; asymptotes \(x=0\), \(R=23\).
(e) \(\frac{\mathrm{d}R}{\mathrm{d}x}=-\frac{5000}{x^2}<0\): unit cost falls as production increases.
(f) 97 bicycles, assuming monthly production is sold.
(b) \(1130\ \mathrm{m}^2\); arrange \(230\ \mathrm{m}^2\) extra storage or stagger deliveries.
(c) 73 dollars per bicycle when 100 are produced per month.
(d) Sketch shown; asymptotes \(x=0\), \(R=23\).
(e) \(\frac{\mathrm{d}R}{\mathrm{d}x}=-\frac{5000}{x^2}<0\): unit cost falls as production increases.
(f) 97 bicycles, assuming monthly production is sold.
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