Hard cone problem

The surface of the cone is 200 cm², its height is 7 centimeters. Calculate the volume of this cone.

Correct answer:

V =  168.4906 cm³

Step-by-step explanation:

$$\begin{gathered} S=200{\text{cm}}^2 \\ h=7\text{ cm } \\ S=\pi \cdot r^2+\pi \cdot r\cdot s \\ {s}^2=r^2+h^2 \\ s=\sqrt{r^2+h^2} \\ S=\pi \cdot r^2+\pi \cdot r\cdot \sqrt{r^2+h^2} \\ r={\displaystyle \frac{S}{\sqrt{\pi }\cdot \sqrt{h^2\pi +2\cdot S}}}={\displaystyle \frac{200}{\sqrt{3.1416}\cdot \sqrt{7^2\cdot 3.1416+2\cdot 200}}}\doteq 4.7943\text{ cm } \\ V={\displaystyle \frac{1}{3}}\cdot \pi \cdot r^2\cdot h={\displaystyle \frac{1}{3}}\cdot 3.1416\cdot 4.7943^2\cdot 7\doteq 168.4906=168.4906{\text{cm}}^3 \\ \text{ Verifying Solution: } \\ s=\sqrt{r^2+h^2}=\sqrt{4.7943^2+7^2}\doteq 8.4844\text{ cm } \\ {S}_2=\pi \cdot r^2+\pi \cdot r\cdot s=3.1416\cdot 4.7943^2+3.1416\cdot 4.7943\cdot 8.4844=200{\text{cm}}^2 \end{gathered}$$

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