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.

Given,

Length of rectangular garden = 10 m

Breadth of rectangular garden = 16 m

Width of walk = x

So, length of garden and walk combined = (10 + x + x) m

Breadth of garden and walk combined = (16 + x + x) m

∴ Area of garden and walk combined = Length $\times$ Breadth = (10 + 2x)(16 + 2x) m²

Given, area of walk = 120m²

Area of walk = Area of combined - Area of garden

$\Rightarrow (10 + 2x)(16 + 2x) - 10 \times 16 = 120 \\[0.5em] \Rightarrow 160 + 20x + 32x + 4x^2 - 160 = 120 \\[0.5em] \Rightarrow 4x^2 + 52x = 120 \\[0.5em] \Rightarrow 4x^2 + 52x - 120 = 0 \\[0.5em] \Rightarrow 4(x^2 + 13x - 30) = 0 \\[0.5em] \Rightarrow x^2 + 13x - 30 = 0 \\[0.5em] \Rightarrow x^2 + 15x - 2x - 30 = 0 \\[0.5em] \Rightarrow x(x + 15) - 2(x + 15) = 0 \\[0.5em] \Rightarrow (x - 2)(x + 15) = 0 \\[0.5em] \Rightarrow x - 2 \text{ or } x + 15 = 0 \\[0.5em] x = 2 \text{ or } x = -15$

Since, width cannot be negative hence x ≠ -15.

The equation in x = (10 + 2x)(16 + 2x) - 10 x 16 = 120.
Value of x = 2m.