Question
A rectangular field is $16\ m$ long and $10\ m$ wide. There is a path of uniform width all around it, having an area of $120\ m^2$. Find the width of the path.
✓
Answer
Let the width of the path be x m Length of the field including the path $= 16 + x + x = 16 + 2x $
Breadth of the field including the path $= 10 + x + x = 10 + 2x$
Now, (Area of the field including path) - (Area of the field excluding path) = Area of the path
$\Rightarrow (16 + 2x)(10 + 2x) - (16 \times 10) = 120 $
$\Rightarrow 160 + 32x + 20x + 4x^2 - 160 = 120 $
$\Rightarrow 4x^2 + 52x - 120 = 0 $
$\Rightarrow x^2 + 13x - 30 = 0 $
$\Rightarrow x^2 + (15 - 2)x + 30 = 0 $
$\Rightarrow x^2 + 15x - 2x + 30 = 0 $
$\Rightarrow x(x + 15) - 2(x + 15) = 0$
$ \Rightarrow (x + 15)(x - 2) = 0$
$\Rightarrow x + 15 = 0$ or $x - 2 = 0$
$\Rightarrow x = -15$ or $x = 2$
$\Rightarrow x = 2$
$[\because$ width cannot be negative$]$Thus, the width of the path is $2m$.
Breadth of the field including the path $= 10 + x + x = 10 + 2x$
Now, (Area of the field including path) - (Area of the field excluding path) = Area of the path
$\Rightarrow (16 + 2x)(10 + 2x) - (16 \times 10) = 120 $
$\Rightarrow 160 + 32x + 20x + 4x^2 - 160 = 120 $
$\Rightarrow 4x^2 + 52x - 120 = 0 $
$\Rightarrow x^2 + 13x - 30 = 0 $
$\Rightarrow x^2 + (15 - 2)x + 30 = 0 $
$\Rightarrow x^2 + 15x - 2x + 30 = 0 $
$\Rightarrow x(x + 15) - 2(x + 15) = 0$
$ \Rightarrow (x + 15)(x - 2) = 0$
$\Rightarrow x + 15 = 0$ or $x - 2 = 0$
$\Rightarrow x = -15$ or $x = 2$
$\Rightarrow x = 2$
$[\because$ width cannot be negative$]$Thus, the width of the path is $2m$.
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