For anyone curious, the web2.0calc post is here
https://web2.0calc.com/questions/a-molasses
where the user CPhill has posted his solution.
This is the original diagram
and I'll add these labels
x = height of the larger cylinder (figure 1) such that
14-x+y = height of the smaller cylinder
V1 = volume of the larger cylinder (figure 1)
V2 = volume of the extra molasses (figure 1)
V3 = volume of the smaller cylinder (figure 2)
V4 = volume of the extra molasses (figure 2)
Each volume (V1 through V4) represents a cylinder of some kind.
V2 and V4 are part of the top cylinder for each respective figure.
Notice in figure 2, I show the calculation of:
58/3 - (14-x+y) = x-y+16/3
The 58/3 is the improper fraction form of the mixed number 19 & 1/3, aka 19 + 1/3
Subtracting off the small cylinder's height, 14-x+y, will get us the remaining height of the molasses for figure 2. Then we have z to finish it off.
Volume of a cylinder = pi*(radius)^2*(height)
V1 = pi*r^2*h
V1 = pi*6^2*x
V1 = 36pi*x
V2 = pi*r^2*h
V2 = pi*2^2*(14-x)
V2 = 4pi*(14-x)
V3 = pi*r^2*h
V3 = pi*2^2*(14-x+y)
V3 = 4pi*(14-x+y)
V4 = pi*r^2*h
V4 = pi*6^2*(x-y+16/3)
V4 = 36pi*(x-y+16/3)
Assuming there aren't any leaks in the container, we can say this:
V1+V2 = V3+V4
which basically says rearranging the contents of the container will not change the overall volume.
So,
V1+V2 = V3+V4
36pi*x+4pi*(14-x) = 4pi*(14-x+y)+36pi*(x-y+16/3)
9x+(14-x) = (14-x+y)+9(x-y+16/3)
The jump from step 2 to step 3 involved dividing both sides by 4pi.
Conveniently all of the pi terms cancel, and things get a bit more simple.
Now we don't need to find x, and we only need the value of y.
This way we can compute 14+y later on (see figure 1).
As the steps below will indicate, the x values cancel out and we can solve for y as such:
9x+(14-x) = (14-x+y)+9(x-y+16/3)
9x+14-x = 14-x+y+9x-9y+48
8x+14 = 14+8x-8y+48
8x+14-8x-14 = -8y+48
0 = -8y+48
8y = 48
y = 48/8
y = 6
Then,
total height = 14+y = 14+6 = 20 cm
when referencing figure 1
It's not a coincidence we get the same solution CPhill did on the link I posted above.
This is because the x terms cancel out.
So for the sake of simplicity, we could have x = 14.
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Answer: 20 cm