A path of width 3.5 m runs around a semi­-circular grassy plot whose perimeter is 72 m. Find the area of the path. (Use $\pi = \frac{22}{7}$)

Given:

A path of width 3.5 m runs around a semi­-circular grassy plot whose perimeter is 72 m.

To do:

We have to find the area of the path.

Solution:

Width of the path around the semicircular grassy plot $= 3.5\ m$.

Perimeter of the plot $= 72\ m$

Let $r$ be the radius of the plot. 

This implies,

$\frac{2\pi r}{2}+2r=72$

$\Rightarrow \frac{22}{7} r+2r=72$

$\Rightarrow \frac{36}{7}r=72$

$\Rightarrow r=72\times \frac{7}{36}$

$\Rightarrow r=14$

The radius of the plot is $14 \mathrm{~m}$.

Width of the path $=3.5 \mathrm{~m}$

Outer radius $R=14+3.5=17.5 \mathrm{~m}$

Area of the path $=\frac{1}{2} \pi(\mathrm{R}^{2}-r^{2})$

$=\frac{1}{2} \times \frac{22}{7}(17.5^{2}-14^{2}) \mathrm{m}^{2}$

$=\frac{11}{7}(306.25-196) \mathrm{m}^{2}$

$=\frac{11}{7} \times 110.25$

$=173.25 \mathrm{~m}^{2}$

The area of the path is $173.25\ m^2$.

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Updated on: 10-Oct-2022

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