An inverted cone of radius a and height h is filled with water. A sphere made of a material denser than water is placed in the cone. Find the radius of the sphere, r, that will displace the largest volume of water.
(In reply to
solution by Steven Lord)
Very nice work Steven. You beat me by 1 1/2 hours. I choose to use the 2D analogy outlined in previous posts and reduced the problem to maximizing the area of a circle of radius r, contained (fully or partially) in an isosceles triangle of base 2a and height h.
Given my very rusty calculus, I used a spreadsheet with parameters for a and h, which also allowed calculation of confirming factors such as chord length to check the calculations.
In the end, I found the exact same transition points (12, 23, 34) as Steven for his two given cases. However, I found slightly different answers for the maximum. 2.25 vs. Steven's 2.55 and 1.626 vs. Steven's 1.73.
Also, I explored the limits of a vs. h a bit. for large h/a, the maximum tends to the transition 12 point, which makes sense. I suspect that for very small h/a, the solution tends towards the 34 transition, but the trend is much slower and this statement is more difficult to support as I keep ruining into resolution issues with Excel.

Posted by Kenny M
on 20190926 07:01:18 