In the picture given below, one can see a rectangular in-ground swimming pool installed by a family in their backyard. There is a concrete sidewalk around the pool of width $x$ m. The outside edges of the sidewalk measure 7 m and 12 m. The area of the pool is 36 sq. m.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding stimulus
Model Answer
(a) Forming the quadratic equation:
The outside edges of the sidewalk are 7 m and 12 m.
Since the sidewalk has width $x$ m on each side, the dimensions of the pool are:
- Length = $(12 - 2x)$ m
- Breadth = $(7 - 2x)$ m
Area of pool = 36 sq. m
$$\therefore (12 - 2x)(7 - 2x) = 36$$
$$84 - 24x - 14x + 4x^2 = 36$$
$$4x^2 - 38x + 84 - 36 = 0$$
$$4x^2 - 38x + 48 = 0$$
$$\boxed{2x^2 - 19x + 24 = 0}$$
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(b) Finding the width of the sidewalk:
Solving $2x^2 - 19x + 24 = 0$:
$$x = \frac{19 \pm \sqrt{361 - 192}}{4} = \frac{19 \pm \sqrt{169}}{4} = \frac{19 \pm 13}{4}$$
$$x = \frac{32}{4} = 8 \quad \text{or} \quad x = \frac{6}{4} = 1.5$$
Since $x = 8$ is not possible (pool dimension would be negative), we reject it.
$$\therefore x = 1.5 \text{ m}$$
The width of the sidewalk is 1.5 m.
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Explanation
- In part (a), subtract $2x$ from both dimensions (sidewalk exists on both sides), expand, and simplify to standard form $ax^2 + bx + c = 0$.
- In part (b), use the quadratic formula and reject the extraneous root ($x = 8$ makes pool dimensions negative). Always verify feasibility — examiners award a mark for explicitly rejecting the invalid root.