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A Frosty Puddle
Age 16 to 18
Challenge Level 
This problem follows on from Frosty the Snowman so the results from this problem will be useful here.
Why do you think the first request asks you to consider the range $2R<h \le 10R$?
Try equating your expression for $h$ to $2R$ and finding $kt$ in terms of $R$. Use this to find the values of $r_1$ and $r_2$ at this point.
In the range $2R < h \le 10R$, can you express $V$ in terms of $h$?
The equation for the volume of a sphere is given at the top of this page.
Can you differentiate your expression for $V$ to find $\dfrac{\mathrm{d} V}{\mathrm{d} h}$?
You may find it easier to expand before differentiating, or you might like to use the chain rule to differentiate the cube terms.
What can you say about the situation when $h<2R$? Can you express $h$ in terms of $t$? Can you use this to express $V$ in terms of $h$ and differentiate?
Think about your values of $r_1$ and $r_2$ when $h=2R$, and what this tells you about Frosty when $h<2R$.
The sketch of $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ will be in two parts. You will need to show clearly what happens at the $h=2R$ boundary.
It is a good idea to calculate the values of $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ when $h=0, 2R, 4R$. You should calculate the value when $h=2R$ using both of the formulae for $\dfrac{\mathrm{d} V}{\mathrm{d} h}$.
How are the graphs of $V$ and $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ related?
You could think about $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ being the gradient of $V$, or consider the fact that the graph of $V$ shows the area under $\dfrac{\mathrm{d} V}{\mathrm{d} h}$.