Rotary cone

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

Correct answer:

S =  275.27 cm²

Step-by-step explanation:

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

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