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A path of 4 m width runs round a semicircular grassy plot whose circumference is $163\frac{3}{7}\ m$. Find the area of the path.
Given:
A path of 4 m width runs round a semicircular grassy plot whose circumference is $163\frac{3}{7}\ m$.
To do:
We have to find the area of the path.
Solution:
Width of the path around the semicircular grassy plot $= 4\ m$.
Circumference of the plot $= 81\frac{5}{7}\ m$
$=\frac{572}{7}\ m$
Let $r$ be the radius of the plot.
This implies,
$\frac{2\pi r}{2}=\frac{572}{7}$
$\Rightarrow \frac{22}{7} r=\frac{572}{7}$
$\Rightarrow r=\frac{572}{7} \times \frac{7}{22}$
$\Rightarrow r=26$
The radius of the plot is $26 \mathrm{~m}$.
Width of the path $=4 \mathrm{~m}$
Outer radius $R=26+4=30 \mathrm{~m}$
Area of the path $=\frac{1}{2} \pi(\mathrm{R}^{2}-r^{2})$
$=\frac{1}{2} \times \frac{22}{7}(30^{2}-26^{2}) \mathrm{m}^{2}$
$=\frac{11}{7}(900-676) \mathrm{m}^{2}$
$=\frac{11}{7} \times 224$
$=352 \mathrm{~m}^{2}$
The area of the path is $352\ m^2$.