163707

First of all you find the area of the top sphere originally, using the sphere equation, (4/3 pie x radius cubed) which is 32/3 pie. Then you find the area of the bottom sphere originally, using the sphere equation which is 36 pie. Then you add them together to find the original volume which is 140/3 pie.
Next, you work out the original height which is 5. Then half of that original height is 2.5. However both the top sphere and the bottom sphere’s heights decrease at the same rate meaning when it is half of its height, the bottom sphere has a radius of 1.75 and the top sphere has a radius of 0.75.
Using these numbers, you can work out the volume of frosty the snowman when he is half of his height. The top sphere has a volume of 9/16 pie when its height is 0.75, using the sphere equation, however the bottom sphere has a volume of 343/48 pie when its height is 1.75. Adding these together we get the number 185/24 pie.
If we divide the volume of frosty when he’s half his height by the volume of frosty when he’s his full height we get the ratio stated in the question, 37:224.
When frosty is a tenth of his original height, we know that his height will be 0.5. However both of the spheres decrease at the same rate meaning that the top sphere doesn’t exist anymore and the bottom sphere will have a radius of 0.5.
Using this, the only sphere in the picture will have a radius of 0.5. This means that the area of this sphere using the sphere equation stated above, is 1/8 pie. This is the full height of frosty as there is no other sphere.
Using a similar method to before, we divide 1/8 pie by the original volume of 140/3 which gets us the ratio of 3:1120. This is the answer to the second question.