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 Partially filled cone (Posted on 2023-02-01)
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?

 No Solution Yet Submitted by Jer No Rating

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 solution | Comment 2 of 3 |
The volume of a cone is h*A/3 where A is the area of the base.

Let h1 be the 8 cm height of air near the tip of the cone.

Let h3 be the full height of the cone.

Let h2 be the h3-2 cm filled with water when the point is held downward.

To avoid complications, just consider the fact that the three cones involved are all similar so the volume is proportional to the cube of the height. The area of the base is proportional to the square of the height and thus the volume is proportional to the cube of the height.

8^3 = h3^3 - h2^3

h3 - h2 = 2

While these are arbitrary units as regard volume, the 8 and h3 and h2 are all in centimeters, so the solution of these simultatneous equations should give us the answer in centimeters.

Calling h3, c; and referring to h2 as b; asking WolframAlpha

solve c^3-b^3=8^3, c-b=2 for c, b

gives h3 = c = 1 + sqrt(85) ~= 10.2195444572929
h2 = b = sqrt(85) - 1 ~=  8.21954445729289

or exactly  1 + sqrt(85) cm.

 Posted by Charlie on 2023-02-01 09:32:56

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