A Frosty Puddle

Frosty the snowman is made from two uniform spherical snowballs, of initial radii $2R$ and $3R.$ The smaller (which is his head) stands on top of the larger. As each snowball melts, its volume decreases at a rate which is directly proportional to its surface area, the constant of proportionality being the same for both snowballs. During melting each snowball remains spherical and uniform. When Frosty is half his initial height, find the ratio of his volume to his initial volume.

If $V$ and $S$ denote his total volume and surface area respectively, find the maximum value of $\dfrac{\mathrm{d}V}{\mathrm{d}S}$ up to the moment when his head disappears.

Frosty the snowman is made from two uniform spherical snowballs, of radii $2R$ and $3R.$ The smaller (which is his head) stands on top of the larger. As each snowball melts, its volume decreases at a rate which is directly proportional to its surface area, the constant of proportionality being the same for both snowballs. During melting, the snowballs remain spherical and uniform. When Frosty is half his initial height, show that the ratio of his volume to his initial volume is 37 : 224.

Let $V$ and $h$ denote Frosty's total volume and height at time $t$. Show that, for $2R <h \le 10R$, $$\dfrac{\mathrm{d} V}{\mathrm{d} h}=\frac{\pi} 8 (h^2 + 4R^2)$$

and derive the corresponding expression for $0 \le h < 2R$.

Sketch $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ as a function of $h$ for $4R \ge h \ge 0$.  Hence give a rough sketch of $V$ as a function of $h$.

Frosty the snowman is made from two uniform spherical snowballs, initially of radii $2R$ and $3R.$ The smaller (which is his head) stands on top of the larger. As each snowball melts, its volume decreases at a rate which is directly proportional to its surface area, the constant of proportionality being the same for each snowball. During melting, the snowballs remain spherical and uniform. When Frosty is half his initial height, show that the ratio of his volume to his initial volume is 37 : 224.

What is this ratio when Frosty is one tenth of his initial height?