Question

Find the volume of a cone, if its total surface area is $$7128\mathrm { sq.cm }$$ and radius of base is $$28 \mathrm { cm } . \left( \pi = \dfrac { 22 } { 7 } \right)$$

Answer 1

Let the perpendicular height and the slant height of the cone be $$h\ cm$$ and $$I\ cm$$ respectively
Radius of the base of cone $$r=28\ cm$$
Total surface area of the cone $$=7128\ cm^2$$
$$\therefore \pi r (r+I)=7128\ cm^2$$
$$\Rightarrow \dfrac{22}{7} \times 28\times (28+l)=7128$$
$$\Rightarrow 28+l=\dfrac{7128}{22 \times 4}\\=81$$
$$\Rightarrow I=81-28\\=53\ cm$$
Now,
$$r^2+h^2=I^2$$
$$\Rightarrow (28)^2+h^2=(53)^2$$
$$\Rightarrow h^2=2809-784\\=2025$$
$$\Rightarrow h=\sqrt{2025}\\=45\ cm$$
$$\therefore$$ Volume of the cone is

$$V=\dfrac{1}{3} \pi r^2 h\\=\dfrac{1}{3} \times \dfrac{22}{7} \times (28)^2 \times 45\\=36960\ cm^3$$
Thus, the volume of the cone is $$36960\ cm^3$$