The initial height of the snowman is 2*2R + 2*3R (sum of diameters) which is equal to 10R.
The formula for the volume of a sphere is (4/3)pi r^3 where r is the radius and pi is the constant 3.141592...
To find the initial volume, we simply substitute the values 2R and 3R into the general formula:
(4/3)pi (2R)^3 + (4/3)pi (3R)^3 = (4/3)pi (8(R^3) + 27(R^3)) = (4/3)pi (35(R^3)) = (140/3)pi R^3
a)
5R is half of the initial snowman's height. Let D be the amount each radius decreases by every minute. After x minutes, the top snowball has a radius of 2R-xD and the bottom one has a radius of 3R-xD.
Using these facts, we can create a formula for the height of the snowman after x minutes:
2 * (2R-xD) + 2 * (3R-xD) = 4R-2xD+6R-2xD = 10R-4xD
At the point where the total height is 5R, we get 10R-4xD=5R => 4xD=5R => xD=(5/4)R. By plugging this into the expressions for the top and bottom radii, we get:
Top radius = 2R - (5/4)R = (3/4)R
Bottom radius = 3R - (5/4)R = (7/4)R
As we can see, these radii indeed make a snowman of height 5R, as:
2 * (3/4)R + 2 * (7/4)R = (6/4 + 14/4)R = (20/4)R = 5R
Next, we can find the volumes of the snowballs. To recap, the radii are (3/4)R and (7/4)R.
(4/3)pi (((3/4)R)^3 + ((7/4)R)^3) = (4/3)pi ((27/64 + 343 / 64)(R^3)) = (4/3)pi ((370/64)(R^3)) = (1480/192)pi R^3 = (740/96)pi R^3 = (370/48)pi R^3 = (185/24)pi R^3
Finally, we evaluate the ratio between the new volume and the initial volume. Due to the way division works, we can ignore pi R^3 and focus on the other coefficients: the smaller one is 185/24 and the larger one is 140/3. By getting them to the same denominator, we effectively compare 185/24 and 1120/24.
The ratio is 185 : 1120; simplified, this is 37 : 224 as given in the problem statement.
b) So far, we have built up quite a few formulae; now, we can finally apply them.
The snowman's height is R when the height reaches one-tenth of the initial height, 10R. Remember that the formula for the height after x minutes is 10R-4xD; so 10R-4xD = R => 4xD=9R => xD=(9/4)R. We can now substitute into the formulae for top / bottom radii:
Top radius = 2R - (9/4)R = "(-1/4)R"
Bottom radius = 3R - (9/4)R = (3/4)R
This part seems a little weird. Negative magnitudes are nonexistent. However, we can proceed because the volume of the bottom snowball is greater than the top radius and so the total volume is still positive. Also note that in this scenario, the top snowball has already melted.
The total volume is:
4/3 pi ((-1/64)(R^3) + (27/64)(R^3)) = 4/3 pi ((13/32)(R^3)) = (52/96) pi (R^3) = (13/24) pi (R^3)
As usual, we ignore pi R^3 and focus on the other coefficient. We compare 13/24 and 140/3; by scaling them up to the same denominator, we get 13/24 and 1120/24. The ratio therefore is:
13 : 1120
which cannot be simplified further.