Rotary cone

The volume of the rotation of the cone is 733 cm³. The angle between the side of the cone and the base angle is 75°. Calculate the lateral surface area of this cone.

Final Answer:

S =  397.72 cm²

Step-by-step explanation:

$$\begin{gathered} V=733{\text{cm}}^3 \\ \alpha =75{}^{\circ } \\ V=\frac{1}{3}\pi r^{2}h \\ \tan \alpha =h:r=h/r \\ h=r\tan \alpha \\ V=\frac{1}{3}\pi r^3\tan \alpha \\ r=\sqrt[3]{\frac{3\cdot V}{\pi \cdot \tan \alpha }}=\sqrt[3]{\frac{3\cdot V}{\pi \cdot \tan 75°}}=\sqrt[3]{\frac{3\cdot 733}{3.1416\cdot \tan 75°}}=\sqrt[3]{\frac{3\cdot 733}{3.1416\cdot 3.732051}}=5.72413\text{cm} \\ \cos \alpha =r:s=r/s \\ s=\frac{r}{\cos \alpha }=\frac{r}{\cos 75°}=\frac{5.7241}{\cos 75°}=\frac{5.7241}{0.258819}=22.11633\text{cm} \\ S=\pi \cdot r\cdot s=3.1416\cdot 5.7241\cdot 22.1163\doteq 397.7152{\text{cm}}^2 \\ \text{ Verifying Solution: } \\ h=\sqrt{s^2-r^2}=\sqrt{22.1163^2-5.7241^2}\doteq 21.3627\text{cm} \\ A=\frac{180°}{\pi }\cdot \arccos (r/s)=\frac{180°}{\pi }\cdot \arccos (5.7241/22.1163)=75{}^{\circ } \\ {V}_2=\frac{1}{3}\cdot \pi \cdot r^2\cdot h=\frac{1}{3}\cdot 3.1416\cdot 5.7241^2\cdot 21.3627=733{\text{cm}}^3 \end{gathered}$$

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