There is a square field whose side is 10 m. A square flower bed i… - Vidyadip

Question

There is a square field whose side is 10 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at ₹ 3 and ₹ 4 per square metre respectively is ₹ 364. Find the width of the gravel path

✓

Answer

Let the width of the gravel path be ‘x’
Side of the flower bed = 10 – (x + x)
= 10 – 2x

Area of the path way $=$ Area of the field - Area of the flower bed
$=10 \times 10-(10-2 x)(10-2 x) \text { sq.m }$
$=100-\left(100+4 x^2-40 x\right)$
$=100-100-4 x^2+40 x$
$=40 x-4 x^2 \text { sq.m }$
Area of the flower bed $=(10-2 x)(10-2 x)$ sq.m.
$=100+4 x^2-40 x$
By the given condition

$3\left(100+4 x^2-40 x\right)+4\left(40 x-4 x^2\right)=364 \\
$300+12 x^2-120 x+160 x-16 x^2=364$
$-4 x^2+40 x+300-364=0$
$-4 x^2+40 x-64=0$
$\Rightarrow x^2-10 x+16=0 \ldots(\div \text { by } 4)$
[The width must not be equal to 8 m since the side of the field is 10 m ]
$(x-8)(x-2)=0 $
$x-8=0 \text { or } x-2=0 $
$x=8 \text { or } x=2$
Width of the gravel path $=2 m$

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