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 $a$?

Using the Sine Law, i.e.
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k $$
we find that the condition in the question is merely equivalent to $k^3 = 7$.
So $a = k \sin A \le k = \sqrt[3]7$.
