Question

$ABCD$ is a square of side $14$ cm and $AED$ and $BEC$ are semi-circles as shown in the figure. Find the area of the shaded region.

Answer 1

Let the radii and height of cone and cylinder be round h.

$\frac{CSAofcylinder}{CSAofcone}=\frac{8}{5}\Rightarrow \frac{2\pi rh}{\pi rl}=\frac{8}{5}\Rightarrow \frac{2h}{l}=\frac{8}{5}$

squaring both sides

$\frac{4h^2}{t^2}=\frac{64}{25}$

Since $l^2=r^2+h^2\Rightarrow \frac{4h^2}{r^2+h^2}=\frac{64}{25}$

$\Rightarrow 64(r^2+h^2)=25\times 4h^2$

$\Rightarrow 16r^2+16h^2=25h^2$

$\Rightarrow 16r^2=25h^2-16h^2=9h^2$

$\Rightarrow (4r{)}^2=(3h{)}^2$

$\Rightarrow 4r=3h$

$\Rightarrow \frac{r}{h}=\frac{3}{4}\therefore r:h=3:4$