1. a semi circular garden has 7 m wide path around it. the perimeterof the garden is 360m find the are of the path

Answer :

Let radius of semicircular garden  =  r m
And
Width of path around 7 m  , So  we form our diagram from given information , As :

Also given Perimeter of garden  =  360 m 

And we know perimeter of semicircle  = $\pi$r  + 2r 
So,

$$\begin{gathered} \Rightarrow \pi r + 2 r = 360 \\ \Rightarrow \frac{22 r}{7}+ 2 r = 360 ( As we know \pi = \frac{22}{7} ) \\ \Rightarrow \frac{22 r + 14r}{7} = 360 \\ \Rightarrow 36r = 360 \times 7 \\ \Rightarrow r =10 \times 7 \\ \Rightarrow r =70 m \end{gathered}$$

So,
Radius of garden with path  =  70 + 7 =  77 m 

And

We know Area of semicircle  = $\frac{\pi r^2}{2}$
So,
Area of Path  = $\frac{{\displaystyle \frac{22}{7}}\times 77\times 77}{2} - \frac{{\displaystyle \frac{22}{7}\times 70\times 70}}{2} = \frac{{\displaystyle 22\times 11\times 77}}{2} - \frac{{\displaystyle 22\times 10\times 70}}{2} = 11\times 11\times 77 - 11\times 10\times 77 = 9317 - 8470 =\mathbf{ }\mathbf{847}\mathbf{ }{\mathit{m}}^{\mathbf{2}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathbf{ }\mathbf{)}$