# 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:

$V = \dfrac13 \pi r^2 h = 472 \ cm^3 \ \\ \tan 70 ^\circ = \dfrac{ h}{r} \ \\ h = r \tan 70 ^\circ \ \\ V = \dfrac13 \pi r^3 \tan 70 ^\circ \ \\ r = \sqrt[3]{\dfrac{ 472 }{\pi \tan 70 ^\circ }} = 5.47 \ cm \ \\ \ \\ \cos 70 ^\circ = \dfrac{r}{s} \ \\ s = \dfrac{ r}{ \cos 70 ^\circ } = 16.01\ cm \ \\ S = \pi r s = 275.27 \ \text{cm}^2$

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