Triangle and Maxium value Given any triangle ABC with $a \ge b \ge c$ such that $\frac{a^3+b^3+c^3}{\sin^3(A)+\sin^3(B)+\sin^3(C)}=7$, what is the maximum value of $a$?

since 
$$\sin^3{A}+\sin^3{B}+\sin^3{C}=(8R^3)^{-1}(a^3+b^3+c^3)$$
so
$$(8R^3)^{-}=\dfrac{1}{7}\Longrightarrow R^{-1}=\sqrt[3]{\dfrac{1}{56}}$$
so
$$a=2R\sin{A}\le 2R=\sqrt[3]{7}$$
