Originally posted by RioXingu
[...]I'm still working on bow-fronts.[...]
Awesome! Even though you left it as an exercise for the student you couldn't resist the urge to be prepared to check their work. Have you considered teaching? :laugh: I wish some of my Geometry students would take up the aquarium hobby, and be stuck buying hex tanks with no listed capacity, maybe it would help them to find some relevance for their studies.
Since you're still working on bow-fronts, may I propose a couple of approaches to the problem?
I think we can safely assume that the bow shape is an elliptical curve, so we can use the area of an ellipse (1/2 * minor axis * 1/2 * major axis * Pi), the wrinkle on this approach is extrapolating the major and minor axes of the ellipse in question from the small part that actually forms the front of the aquarium.
Another possibility, less elegant but
much easier, is to "simplify" the bow front to a triangle. Determining the altitude of said "triangle" requires only a sufficiently long straightedge. Having found the altitude, we now insert the time honored "fudge factor" into our formula. Instead of (1/2 * base * altitude) we nudge the 1/2 upward. I propose 7/12 as it is the median value between two limits that can be established on our fudge factor.
{For the upper limit, I am using an alternate form for the volume of a sphere, namely (2/3 * the volume of the right-circular cylinder that would exactly contain the sphere in question)}
Anything you have to say on this topic will be of interest to me
