It's All Volumes!

Clearly in both cases the volume is the same.

In both diagrams below replace $RQ = x$ by $\dfrac{x}{2}$ and $ST = x_{1}$ and $MN = x_{2}$ by $\dfrac{x_{1}}{2}$ and $\dfrac{x_{2}}{2}$ respectively.

For volume $V_{1}$ :

Using the above diagram $\triangle{PTS} \sim \triangle{PRQ} \implies \dfrac{16}{x_{1}} = \dfrac{2h}{x} \implies x_{1} = \dfrac{8x}{h} \implies$

$V_{1} = \dfrac{1}{3}(x^2 - 8(\dfrac{64x^2}{h^2})) = \dfrac{1}{3}x^2(\dfrac{h^3 - 512}{h^2})$

For $V_{2}$ :

Using the second diagram above $\triangle{P'MN} \sim \triangle{P'RQ} \implies \dfrac{h - 2}{x_{2}} = \dfrac{h}{x}$

$\implies x_{2} = \dfrac{(h - 2)x}{h} \implies V_{2} = \dfrac{1}{3}x^2\dfrac{(h - 2)^3}{h^2}$

$V_{1} = V_{2} \implies h^3 - 512 = h^3 - 6h^2 + 12h - 8 \implies h^2 - 2h - 84 = 0 \implies$

$h = 1 + \sqrt{85}$ choosing the positive root.

$\therefore \boxed{h = 1 + \sqrt{85} \approx 10.2195}$ .