Question

There is a square field whose side is $44m$. A square flower bed is prepared in its centre leaving a gragel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at Rs. $2.75$ and Rs. $1.50$ per square metre, respectively, is Rs. $4904$. Find the width of the gravel path.

Answer 1

Let the width of the gravel path be $x$ metres. Then, Each side of the square flower bed is $(44-2x)$ metres

Now, Area of the square field $44\times 44=1936m^2$

Area of the flower bed $={(44-2x)}^2{m}^2$

Area of the gravel path$=$ Area of the field - Area of the flower bed
$=1936-{(44-2x)}^2=1936-(1936-176x+4x^2)=(176-4x^2)m^2$

Cost of laying the flower bed$=$Area of the flower bed $\times$ Rate per sq.m
$={(44-2x)}^2\times \frac{275}{100}=\frac{11}{4}{(44-2x)}^2=11{(22-x)}^2$

Cost of gravelling the path = Area of the path $\times$ Rate per sq. m
$=(176x-4x^2)\frac{150}{100}=6(44-x^2)$

It is given that the total cost of laying the flower bed and gravelling the path is Rs. $4904$

$\therefore 11{(22-x)}^2+6(44x-x^2)=4904$

$\Rightarrow 5x^2-220x+5324=4908$

$\Rightarrow 5x^2-220x+420=0\Rightarrow x^2-44x+84=0\Rightarrow x^2-42x-2x+84=0$

$\Rightarrow (x-2)(x-42)=0\Rightarrow x=2orx=42$

But $x\ne 42$ as the side of the square is $44m$. therefore, $x=2$

Hence the width of the gravel path is $2$ metres