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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 $= x \operatorname{m}$.
(ii) (a) Solve the quadratic equation to find the width of the walkway 'x'.
OR
(b) If the cost of paving the walkway at the rate of ₹50 per square metre is ₹12,000, calculate the area of the walkway.
(iii) Find the perimeter of the lawn.
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 $= x \operatorname{m}$.
(ii) (a) Solve the quadratic equation to find the width of the walkway 'x'.
OR
(b) If the cost of paving the walkway at the rate of ₹50 per square metre is ₹12,000, calculate the area of the walkway.
(iii) Find the perimeter of the lawn.
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Sol. (i) $(12 + 2x)(10 + 2x) = 360$
$4x^2 + 44x - 240 = 0$ or $x^2 + 11x - 60 = 0$
(ii)(a) $(x + 15)(x - 4) = 0$
$x = 4$
$\therefore$ width of the walkway $= 4 \operatorname{m}$
OR
(ii)(b) Area of the walkway $= \frac{12000}{50}$
$= 240 \operatorname{m}^2$
(iii) Perimeter of the lawn $= 2(12 + 10) = 44 \operatorname{m}$
$4x^2 + 44x - 240 = 0$ or $x^2 + 11x - 60 = 0$
(ii)(a) $(x + 15)(x - 4) = 0$
$x = 4$
$\therefore$ width of the walkway $= 4 \operatorname{m}$
OR
(ii)(b) Area of the walkway $= \frac{12000}{50}$
$= 240 \operatorname{m}^2$
(iii) Perimeter of the lawn $= 2(12 + 10) = 44 \operatorname{m}$