Question

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.

Answer 1

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 m2
Area of the flower bed = xy m2
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 + 4w2 = 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.