The dimensions of a rectangular field are $50 m$ by $40 m$ . A flower bed is prepared inside this field leaving a gravel path of uniform width all round the flower bed. The total cost of laying the flower bed and gravelling the path at $R s 30$ and $R s 20$ per square metre, respectively is $R s 52, 000$ . Find the width of the gravel path.

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Solution

Given,
dimensions of rectangular field $= 50 m \times 40 m$
cost of flowering bed = $30 R s / m^{2}$
cost of grovelling path = $20 R s / m^{2}$

Ref. image
Let width of gravel =x m
area of field = $50 \times 40 = 2000 m^{2}$
area of flower bed = $(50 - 2 x) (40 - 2 x)$

Total cost
$=$ cost of flower bed+ grovelling path
$∴ 52000 = [(50 - 2 x) (40 - 2 x) \times 30]$ $+ [(2 \times 50 \times x) + 2 (40 - 2 x) x] \times 20$
$= 52000 = [(2000 - 100 x - 80 x + 4 x^{2}) \times 30]$ $+ [100 x + 80 x - 4 x^{2}] \times 20$
$= 52000 = 60000 - 5400 x + 120 x^{2} + 3600 x - 8 x^{2}$
$= 40 x^{2} - 1800 x + 8000 = 0$
$= x^{2} - 45 x + 200 = 0$
$∴ x = 5, 40$

$40$ can't be the value of $x$ as
it is the total field width
$∴ x = 5 m$
Thus, gravel path width is $5 m$

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The dimensions of a rectangular field are 50 m by 40 m. A flower bed is prepared inside this field leaving a gravel path of uniform width all around the flower bed. The total cost of laying the flower bed and gravelling the path at Rs. 30 and Rs. 20 per square meter, respectively, is Rs. 52,000. Find width of the gravel path.

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