Q1. [4] § 4.1 Introduction
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.
Based on the information given above, answer the following questions:
- (i) Formulate the quadratic equation representing the total area of the lawn and the walkway, taking width of walkway $= x$ m. [1]
- (ii) Solve the quadratic equation / find the width of the walkway or the area. [2]
- (iii) Find the perimeter of the lawn. [1]
Previously asked in CBSE board exam
2025 30/3/1 Q36
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.
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