Question

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 π = 22/7)
[CBSE 2015]

Answer 1

Let the radius of the semicircular grass plot be r
Now, Perimeter of a semicircular grassy = Circumference of semicircular grassy plot + diameter of semicircular grassy plot
$$\begin{gathered} \Rightarrow 72=\frac{1}{2}\left(2\mathrm{\pi }r\right)+2r \\ \Rightarrow 72=\frac{22}{7}r+2r \\ \Rightarrow 72=\frac{36r}{7} \\ \Rightarrow r=14\mathrm{m} \end{gathered}$$
Now, the distance(R) of the circumference of the path from the centre is given by
14 + 3.5 = 17.5 m
Now, Area of path is given by
$$\begin{gathered} =\frac{1}{2}\mathrm{\pi }{R}^2\mathit{-}\frac{1}{2}\mathrm{\pi }{r}^{\mathit{2}} \\ =\frac{1}{2}\times \frac{22}{7}\times 17.5\times 17.5-\frac{1}{2}\times \frac{22}{7}\times 14\times 14 \\ =481.25-308 \\ =173.25{\mathrm{m}}^2 \end{gathered}$$
=12(2πr)+d=πr+d=πd2+d=227×7+14=22+14=36 cm
Hence, the perimeter of a semicircular protractor is 173.25 m²