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 $352\ m^2$.

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Updated on: 10-Oct-2022

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