Rotary cone

The volume of the rotation of the cone is 472 cm³ and angle between the side of the cone and base angle is 70°. Calculate lateral surface area of this cone.

Correct result:

S =  275.27 cm²

Solution:

$$\begin{gathered} V={\displaystyle \frac{1}{3}}\pi r^{2}h=472c{m}^3 \\ \tan 70^{\circ }={\displaystyle \frac{h}{r}} \\ h=r\tan 70^{\circ } \\ V={\displaystyle \frac{1}{3}}\pi r^3\tan 70^{\circ } \\ r=\sqrt[3]{{\displaystyle \frac{472}{\pi \tan 70^{\circ }}}}=5.47cm \\ \cos 70^{\circ }={\displaystyle \frac{r}{s}} \\ s={\displaystyle \frac{r}{\cos 70^{\circ }}}=16.01cm \\ S=\pi rs=275.27{\text{cm}}^2 \end{gathered}$$

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