Angle of deviation

The surface of the rotating cone is 30 cm² (with circle base), its surface area is 20 cm². Calculate the deviation of the side of this cone from the plane of the base.

Correct result:

A =  60 °

Solution:

$$\begin{gathered} S_1=30{\text{cm}}^2 \\ {S}_2=20{\text{cm}}^2 \\ {S}_1=S_2+\pi r^2 \\ r=\sqrt{(S_1-S_2)\mathrm{/}\pi }=\sqrt{(30-20)\mathrm{/}3.1416}\doteq 1.7841\text{ cm } \\ {S}_2=\pi rs \\ s=S_2\mathrm{/}(\pi \cdot r)=20\mathrm{/}(3.1416\cdot 1.7841)\doteq 3.5682 \\ \cos A=r\mathrm{/}s \\ A={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arccos (r\mathrm{/}s)={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arccos (1.7841\mathrm{/}3.5682)=60^{\circ }=60^{\circ } \end{gathered}$$

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