Higher November 2019 Paper 2 Q9
9. A square lawn has a side of length 8 m.
A builder is asked to make a path around the outside edge of the square lawn.
(a) The plan was for a path of width \(y\) metres with all the edges of the path being straight.
Find an expression for the area of this path in terms of \(y\).
Give your answer in its simplest form. [4]
(b) Before the builder started, the plan was updated and the following agreed:
- at each corner of the lawn the path should be a quarter circle,
- the width of the path is to be 1·5 m,
- the path is to be made of 8 cm thick concrete.
Calculate the volume of the concrete needed for the path. [5]
| Scheme | Marks | Comments |
|---|---|---|
| Initial strategy, e.g. sketch of concentric squares | S1 | For intention, but some sections may be missed |
| Correct method to find area, e.g. subtraction of areas or composite shapes | M1 | |
| \((8+2y)(8+2y) - 8 \times 8\) or \(8y+8y+8y+8y+y^2+y^2+y^2+y^2\) | A1 | Or equivalent |
| \(4y^2 + 32y\) or \(4(y^2 + 8y)\) or \(4y(y + 8)\) or equivalent | A1 | Mark final answer If the path has been built inside then MR-1 and FT |
| Scheme | Marks | Comments |
|---|---|---|
| Strategy, drawing with ¼ circles in corners OR sight of \(\pi \times 1.5^2\) of a fraction or multiple of \(\pi \times 1.5^2\) (or equivalent) | S1 | FT from (a) if possible |
| \(4 \times 8 \times 1.5 + 4 \times \tfrac14 \times \pi \times 1.5^2\) | M1 | Accept if measurements changed to cm consistently throughout |
| \(\times 0.08\) | m2 | m1 for × ‘digit 8’ with incorrect place value |
| 4.4(.... m³) | A1 | CAO If the path has been built inside then MR-1 and FT |
| (9) |