You might find the following useful!
Can you write down a differential equation connecting $V$ and $A$, using what you have been told about the rate at which the volume decreases?
Can you rewrite the differential equation in terms of $r$?
It might be helpful to use notation like $r_1=$ radius of head, $r_2=$ radius of body.
Can you solve the differential equation to find $r_1$ and $r_2$ in terms of $t$?
Can you write the height of Frosty ($h$) as a function of $r_1$ and $r_2$? (it might help to draw a sketch of Frosty!) Can you write $h$ in terms of $t$?
What is the initial value of $h$? Can you find an expression for $t$ when the height is half the initial height?
Can you use this value of $t$ to find out the radii of the two snowballs at this time? Use this to find the volumes of each one.
What is the total volume of Frosty at the start? What is the total volume when the height is half the initial height? Can you find a simplify a ratio between these two total volumes?
At what point does Frosty's head disappear?
In terms of $R$, how tall is Frosty when his height is one tenth of the initial height? What does that tell you?