We are family - 2

Applying the AM/GM inequality $\tfrac13 \; = \; \tfrac13\left(\tfrac{9}{a^2} + \tfrac{16}{b^2} + \tfrac{144}{c^2}\right) \; \ge \; \sqrt[3]{\tfrac{9 \times 16 \times 144}{a^2b^2c^2}} \; = \; \left(\tfrac{144}{abc}\right)^{\frac23}$ so the volume of the ellipsoid is $V \; = \; \tfrac43\pi abc \; \ge \; \tfrac43\pi \times 144 \times 3\sqrt{3} \; = \; \boxed{576\pi\sqrt{3}}$ We can choose $a,b,c$ to obtain equality in the AM/GM inequality, and hence can choose $a,b,c$ to obtain this minimum value of $V$ .

The volume of the ellipsoid becomes minimum when a=3√3, b=4√3 and c=12√3. The minimum volume is $\dfrac{4}{3}$ π(3√3)(4√3)(12√3)=576√3π