[Stacked spheres in a cone]

Place the apex of the cone at the origin and let depth $d$ increase downward along the axis. A sphere of radius $\rho$ centred on the axis at depth $d$ is tangent to the lateral surface iff $\rho = d \sin\alpha$.

For $S_{1}$: $\rho_{1} = 3$, $\sin\alpha = 3/5 \Rightarrow d_{1} = 3 / (3/5) = 5$.

Let $S_{2}$ have radius $\rho_{2}$ centred at depth $d_{2} < d_{1}$. Tangency to lateral surface: $\rho_{2} = d_{2} \sin\alpha = \tfrac{3}{5}d_{2}$. External tangency to $S_{1}$:

$

d_{1} - d_{2} = \rho_{1} + \rho_{2} ;\Longrightarrow; 5 - d_{2} = 3 + \tfrac{3}{5}d_{2}.

$

$

\Rightarrow 2 = \tfrac{8}{5}d_{2} \Rightarrow d_{2} = \tfrac{5}{4},\qquad \rho_{2} = \tfrac{3}{5}\cdot\tfrac{5}{4} = \tfrac{3}{4}.

$

Answer: 0.75 cm.

Why CAT-level: The compact relation $\rho = d \sin\alpha$ is not standard CAT vocabulary; deriving it via the cross-sectional triangle and then using the external-tangency condition is what makes this Very Hard.