A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres.
Generated by claude-sonnet-4-6 · 2026-06-15 10:30 · grounding stimulus
Model Answer
(i) With walkway width = $x$ m, the overall dimensions become $(12 + 2x)$ m by $(10 + 2x)$ m.
Total area: $(12 + 2x)(10 + 2x) = 360$
$120 + 24x + 20x + 4x^2 = 360$
$4x^2 + 44x - 240 = 0$
$$x^2 + 11x - 60 = 0$$
(ii) $x^2 + 11x - 60 = 0$
$(x + 15)(x - 4) = 0$
$x = -15$ or $x = 4$
Since width cannot be negative, $x = 4$ m.
The width of the walkway is 4 metres.
(iii) Perimeter of the lawn = $2(l + b) = 2(12 + 10) = 2 \times 22 = \textbf{44 metres}$
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Explanation
- (i) Add $2x$ to each dimension (walkway on both sides), expand, and simplify to standard quadratic form. Examiners award the mark for the correct equation $x^2 + 11x - 60 = 0$.
- (ii) Factorise or use the quadratic formula; reject the negative root — this step is essential and often carries a dedicated mark.
- (iii) Perimeter uses the lawn dimensions (12 m × 10 m), not the outer boundary. A common mistake is using the total outer dimensions — avoid it.