Question

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

Answer 1

Let the perpendicular height and the slant height of the cone be h cm and l cm, respectively.

Radius of the base of cone, r = 28 cm

Total surface area of the cone = 7128 cm²

∴ $\mathrm{\pi }$r(r + l) = 7128 cm²

$$\begin{gathered} \Rightarrow \frac{22}{7}\times 28\times \left(28+l\right)=7128 \\ \Rightarrow 28+l=\frac{7128}{22\times 4}=81 \\ \Rightarrow l=81-28=53\mathrm{cm} \end{gathered}$$
Now,

$$\begin{gathered} r^2+h^2=l^2 \\ \Rightarrow {\left(28\right)}^2+h^2={\left(53\right)}^2 \\ \Rightarrow h^2=2809-784=2025 \\ \Rightarrow h=\sqrt{2025}=45\mathrm{cm} \end{gathered}$$
∴ Volume of the cone = $\frac{1}{3}\mathrm{\pi }{r}^{2}h\mathit{=}\frac{1}{3}\times \frac{22}{7}\times {\left(28\right)}^2\times 45$ = 36960 cm³

Thus, the volume of the cone is 36960 cm³.