Question

The total surface area of a cone is 71.28 cm². Find the volume of this cone if the diameter of the base is 5.6 cm.

Answer 1

Radius of the cone = $\frac{\mathrm{Diameter}}{2}=\frac{5.6}{2}=2.8\mathrm{cm}$

Total surface area of the cone = $\mathrm{\pi }r(l+r)$

$$\begin{gathered} \Rightarrow 71.28=\mathrm{\pi }r(l+r) \\ \Rightarrow 71.28=\frac{22}{7}\times 2.8(l+2.8) \\ \Rightarrow l+2.8=8.1 \\ \Rightarrow l=8.1-2.8 \\ \Rightarrow l=5.3\mathrm{cm} \end{gathered}$$

We know:
$$\begin{gathered} h=\sqrt{l^2-r^2} \\ =\sqrt{{\left(5.3\right)}^2-{\left(2.8\right)}^2} \\ =\sqrt{28.09-7.84} \\ =\sqrt{20.25} \\ =4.5\mathrm{cm} \end{gathered}$$

Thus, we have:
$$\begin{gathered} \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone}=\frac{1}{3}\mathrm{\pi }{r}^{2}h \\ \mathit{=}\frac{1}{3}\times \frac{22}{7}\times {\left(2.8\right)}^2\times 4.5 \\ =36.96{\mathrm{cm}}^3 \end{gathered}$$