1. Surface area constraint

For the cuboid with sides $2x,4x,5x$: $S_1=2(lb+bh+hl)=2(2x\cdot 4x+4x\cdot 5x+5x\cdot 2x)$ $=2(8x^2+20x^2+10x^2)=76x^2$

For a closed hemisphere of radius $r$, total surface area is: $S_2=3\pi r^2$

Given their sum is constant $k$: $76x^2+3\pi r^2=k \quad ...(1)$

2. Volume expression

Volume of the cuboid: $V_1=(2x)(4x)(5x)=40x^3$

Volume of the hemisphere: $V_2=\frac{2}{3}\pi r^3$

So total volume is: $V=40x^3+\frac{2}{3}\pi r^3$

We must maximize $V$ subject to constraint (1).

3. Use Lagrange multiplier / differentiation under constraint

Let $F=40x^3+\frac{2}{3}\pi r^3-\lambda(76x^2+3\pi r^2-k)$

Then, $\frac{\partial F}{\partial x}=120x^2-152\lambda x=0$ $x(120x-152\lambda)=0$ Since $x>0$, $120x=152\lambda$ $\lambda=\frac{15x}{19}$

Also, $\frac{\partial F}{\partial r}=2\pi r^2-6\pi \lambda r=0$ $2\pi r(r-3\lambda)=0$ Since $r>0$, $r=3\lambda$

Substitute $\lambda=\frac{15x}{19}$: $r=3\cdot \frac{15x}{19}=\frac{45x}{19}$

Hence, $x:r = x : \frac{45x}{19} = 19:45$

4. Check with options

Thus the required ratio is: $\boxed{19:45}$ So the correct option is B.

5. Comparison with stored answer

Stored correct answer = B

Our derived answer = B

Hence, they agree.