All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Partially filled cone (Posted on 2023-02-01) Difficulty: 3 of 5
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

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution 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
      
Answer: about 10.2195444572929 cm      
    or exactly  1 + sqrt(85) cm.

  Posted by Charlie on 2023-02-01 09:32:56
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (8)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information