Imagine a sealed clear hollow cone, partially filled with water.

When the cone is held point down, the water fills all but the top 2cm.
When it is held point up, there is 8 cm of air at the top.

What is the height of the cone?

Cone has radius R and height H
Cone volume = (1/3) pi*R^2 * H
The cone of water and the cone of air have the same ratio between radius and height as the whole cone.
The height of the cone of water is H-2, so the ratio of both radius and height of this cone is (H-2)/H.
The height of the cone of air is 8, so the ratio of both radius and height of this cone is 8/H.

The volumes of the water and air cones are just the volume of the whole cone times the cube of the appropriate ratio.
The whole cone volume is the air volume plus the water volume.
The pi terms and 1/3 terms are common to all 3 cones, so they will all cancel out.  The only equation we need is:

1 = ((H-2)/H)^3 + (8/H)^3
H^3 = H^3 - 6H^2 + 12H - 8 + 512
6H^2 - 12H - 504 = 0
H^2 - 2H - 84 = 0
no need for +/-, since the - version will be negative
H = (2 + sqrt(340))/2
H = 1 + sqrt(85)  = approx 10.2195

Posted by Larry on 2023-02-01 09:08:15