Question

In the adjoining figure, ABCD is a square grassy lawn of area 729 $m^2$. A path of uniform width runs all around it. If the area of the path is 295 $m^2$, find
(i) the length of the boundary of the square field enclosing the lawn and the path.
(ii) the width of the path.

Answer 1

Given

Area of square $ABCD=729m^2$

So, its side $=\sqrt{729}=27m$

Let's take the width of path $=x$ m

Then

Side of out field $=27+x+x=(27+2x)$

and area of square $PQRS={(27+2x)}^2{m}^2$

Now

Area of PQRS - Area of ABCD = Area of path

$\Rightarrow {(27+2x)}^2{m}^2-729m^2=295m^2$

$\Rightarrow 729+4x^2+108x-729=295$

$\Rightarrow 4x^2+108x-295=0$

By using the quadratic formula, we have

$a=4,b=108,c=-295$

$\Rightarrow x=\frac{-108\pm \sqrt{{\left(108\right)}^2-4\times \left(4\right)\times (-295)}}{8}=\frac{-108\pm \sqrt{11664+4720}}{8}=\frac{-108\pm 128}{8}=\frac{20}{8}=2.5$

Hence, Width of the path $=2.5m$

Now side of square field $PQRS=27+2x=(27+2\times 2.5)m=32m$

Therefore,

Length of boundary $=4\times side=32\times 4=128m$