Mathematics
A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 square metres, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x.
Mensuration
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Answer
Let ABCD be a rectangular garden.
Length = 10 m and Breadth = 16 m.
Area of garden ABCD = l × b
= 10 × 16 = 160 m2
Given, width of the walk = x meters.
From figure,
Length of rectangular garden PQRS = 10 + x + x = (10 + 2x) m
Breadth of rectangular garden PQRS = 16 + x + x = (16 + 2x) m
From figure,
⇒ Area of walk = Area of rectangle PQRS - Area of rectangle ABCD
⇒ 120 = (10 + 2x)(16 + 2x) - 160
⇒ 120 = 160 + 20x + 32x + 4x2 - 160
⇒ 120 = 4x2 + 52x
⇒ 4x2 + 52x - 120 = 0
⇒ 4(x2 + 13x - 30) = 0
⇒ x2 + 13x - 30 = 0
The above equation is the equation in x.
Solving further,
⇒ x2 + 15x - 2x - 30 = 0
⇒ x(x + 15) - 2(x + 15) = 0
⇒ (x - 2)(x + 15) = 0
⇒ x = 2 or x = -15.
Since, length cannot be negative.
∴ x = 2.
Hence, equation is x2 + 13x - 30 = 0 and x = 2 metres.
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