Quote (PiXi31415 @ Dec 11 2022 01:42am)
that is correct if we set the geometric theorem that the ratio volume:surface increases the more regular a body gets to be known. my question is though: can that be proven by geometry (without basic calculus or Lagrange optimization or similar means)? or do we just accept it, because it seems obvious?
EDIT: not that it matters for this task, just curious :D
Oof. There might be some roundabout proof, but none that I want to type on my phone nor that a student in this class would pay attention to long enough for it to make any sense.
Most instructors in these lower courses would use some type of self-discovery question to type to lead students to convince students that the fact is true.
You can show that a square is the most optimal rectangle (I would do this through pictures for lower classes)
This then would force us to think that the base of our prism should probably be a square. Then we have to figure out the height.
Choose a constant for a given surface area (anything positive should do).
Rewrite both equations for surface area and volume now that we know it should have a square base.
Solve the surface area formula for height.
Substitute that horrid expression into volume.
You should be a left with a cubic
Graph the cubic equation
The maximum should show the height must be equal to length of the side.
I am not sure I would ever do this in a lower class. Most students tune me out if I try to go over a proof. In my experience you end up discussing the problem with the one brilliant student while the other 29 zone out and feel dumb.