The dimensions of a rectangular field are 50 m and 40 m. A flower… - Vidyadip

Question

The dimensions of a rectangular field are $50 m$ and $40 m$. 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 metre, respectively, is Rs $52,000$. Find the width of the gravel path.

✓

Answer

Let the width of the gravel path be w $m.$
Length of the rectangular field $= 50 m$
Breadth of the rectangular field $= 40 m$
Let the length and breadth of the flower bed be x m and y m respectively.
Therefore, we have:
$x + 2w = 50 … (1)$
$y + 2w = 40 … (2)$
Also, area of rectangular field $= 50 m 40 m = 2000 m^2$
Area of the flower bed $= xy m^2$
Area of gravel path = Area of rectangular field – Area of flower bed $= (2000 – xy) m^2$
Cost of laying flower bed + Gravel path = Area x cost of laying per sq. m
$52000 = 30 xy + 20 (2000 – xy)$
$52000 = 10xy + 40000$
$xy = 1200$
Using (1) and (2), we have:
$(50 – 2w) (40 – 2w) = 1200$
$2000 – 180w + 4w^2 = 1200$
$4w^2 – 180w + 800 = 0$
$w^2 – 45w + 200 = 0$
$w^2 – 5w – 40w + 200 = 0$
$w(w – 5) – 40(w – 5) = 0$
$(w – 5) (w – 40) = 0$
$w = 5, 40$
If $w = 40$, then $x = 50 – 2w = -30$, which is not possible.
Thus, the width of the gravel path is $5 \ m.$

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