A rectangular playing surface is to be constructed so that the length is 6 metres more than the width.
- Give an example of a length and width that would be possible for this playing surface. (1 mark)
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- Write an equation for the area (`A`) of the playing surface in terms of its length (`l`). (1 mark)
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A graph comparing the area of the playing surface to its length is shown.
- Why are lengths of 0 metres to 6 metres impossible? (1 mark)
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- What would be the dimensions of the playing surface if it had an area of 135 m²? (2 marks)
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Company `A` constructs playing surfaces.
- Draw a graph to represent the cost of using Company `A` to construct all playing surface sizes up to and including 200 m².
Use the horizontal axis to represent the area and the vertical axis to represent the cost. (2 marks)
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- Company `B` charges a rate of $360 per square metre regardless of size.
- Which company would charge less to construct a playing surface with an area of 135 m²
Justify your answer with suitable calculations. (1 mark)
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- `text(One possibility is a length of 10 m, and a width)`
`text{of 4 m (among many possibilities).}`
- `A = l (l – 6)\ \ text(m²)`
- `text(Given the length must be 6 m more than)`
`text(the width, it follows that the length)`
`text(must be greater than 6 m.)`
- `text(15 m × 9 m)`
-
- `text(Proof)\ \ text{(See worked solutions)}`
i. `text(One possibility is a length of 10 m, and a width)`
`text{of 4 m (among many possibilities).}`
ii. `text(Length) = l\ text(m)`
`text(Width) = (l – 6)\ text(m)`
| `:.\ A` | `= l (l – 6)` |
iii. `text(Given the length must be 6m more than the width,)`
`text(it follows that the length must be greater than 6 m)`
`text(so that the width is positive.)`
iv. `text(From the graph, an area of 135 m² corresponds to)`
`text(a length of 15 m.)`
`:.\ text(The dimensions would be 15 m × 9 m.)`
| v. |  |
vi. `text(Company)\ A\ text(cost) = $50\ 000`
| `text(Company)\ B\ text(cost)` | `= 135 xx 360` |
| `= $48\ 600` |
`:.\ text(Company)\ B\ text(would charge $1400 less)`
`text(than Company)\ A.`
