本帖最后由 青青子衿 于 2019-6-10 18:31 编辑
\begin{gather*}
\left\{
\begin{split}
\left(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\right)^2
&\leqslant\dfrac{x}{A}+\dfrac{y}{B}+\dfrac{z}{C}\\
x&\geqslant0\\
y&\geqslant0\\
z&\geqslant0
\end{split}\right.\\
\\
V=\dfrac{abc}{60}\left(\left(\dfrac{a}{A}+\dfrac{b}{B}\right)\left(\dfrac{a^2}{A^2}+\dfrac{b^2}{B^2}\right)+\left(\dfrac{a^2}{A^2}+\dfrac{ab}{AB}+\dfrac{b^2}{B^2}\right)\dfrac{c}{C}+\left(\dfrac{a}{A}+\dfrac{b}{B}\right)\dfrac{c^2}{C^2}+\dfrac{c^3}{C^3}\right)\\
\\
V=\dfrac{abc}{60}\bigg(\left(\dfrac{a}{A}+\dfrac{b}{B}+\dfrac{c}{C}\right)^3-2\left(\dfrac{a}{A}+\dfrac{b}{B}+\dfrac{c}{C}\right)\left(\dfrac{a\,b}{AB}+\dfrac{b\,c}{BC}+\dfrac{a\,c}{AC}\right)+\dfrac{a\,b\,c}{ABC}\bigg)
\end{gather*} |