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 ₹ 30 and ₹ 20 per square meter, respectively, is ₹ 52000. Find the width of the gravel path.

Let the width of gravel path be w meter.

Length of rectangular field = 50 m

Breadth of rectangular field = 40 m

Hence,

Length of flower bed (x) = 50 - 2w ……(i)

Breadth of flower bed (y) = 40 - 2w ……(ii)

Area of flower bed = xy m²

Area of rectangular field = (50)(40) = 2000 m²

Given, cost of laying flower bed + gravel path = 52000.

⇒ 52000 = 30xy + 20(2000 - xy)

⇒ 52000 = 30xy + 40000 - 20xy

⇒ 52000 - 40000 = 10xy

⇒ 12000 = 10xy

⇒ xy = 1200

⇒ From (i) and (ii) we get,

⇒ (50 - 2w)(40 - 2w) = 1200

⇒ 2000 - 100w - 80w + 4w² = 1200

⇒ 4w² - 180w + 2000 - 1200 = 0

⇒ 4w² - 180w + 800 = 0

⇒ 4(w² - 45w + 200) = 0

⇒ w² - 45w + 200 = 0

⇒ w² - 40w - 5w + 200 = 0

⇒ w(w - 40) - 5(w - 40) = 0

⇒ (w - 5)(w - 40) = 0

⇒ w - 5 = 0 or w - 40 = 0

⇒ w = 5 or w = 40.

If w = 40, then x = 50 - 2w = 50 - 2(40) = -30 which is not possible.

Hence, width of the gravel path is 5 m.