+26367 The base diameter of a right circular cone is \(18~\text{cm}\).
If the lateral area is \(516.4~\text{cm}^2\), find its volume in cubic centimeters
\(\text{Let the radius of the circle $ r = 9~\text{cm} $ } \\ \text{Let the lateral area $ F_\text{lateral} = 516.4~\text{cm}^2$ } \\ \text{Let the circle area $ F_\text{circle} = \pi r^2 =81\pi= 254.469\ldots~\text{cm}^2$ }\)
volume:
\(\begin{array}{|rcll|} \hline \mathbf{V_{\text{circular cone}}} &=& \mathbf{\dfrac{r}{3}*\sqrt{F_\text{lateral}^2 - F_\text{circle}^2}} \\\\ &=& \dfrac{9~\text{cm}}{3}*\sqrt{\left(516.4~\text{cm}^2\right)^2 - \left(254.469\ldots~\text{cm}^2\right)^2} \\\\ &=& 3~\text{cm}*\sqrt{266668.96~\text{cm}^4 - 64,754.4744755~\text{cm}^4} \\ &=& 3~\text{cm}*\sqrt{201,914.485524~\text{cm}^4} \\ &=& 3~\text{cm}*449.348957409~\text{cm}^2 \\ &=& 1348.04687223~\text{cm}^3 \\ \mathbf{ \mathbf{V_{\text{circular cone}}} } &=& \mathbf{1348.0~\text{cm}^3} \\ \hline \end{array} \)
