How would one solve questions like this one here in general? I have gotten an answer for that question, but I don't understand what's the intuition behind it. Can smoeone clearly explain how to minimize and maximize the area of triangles (as asked in the linked question) and the intuition behind it?
[Math] Minimizing (and maximizing) the area of triangles
algebra-precalculuscalculusminimal-surfacestriangles
Related Solutions
Your conclusion that $\triangle PVY$ has area $40$ is correct, and your reasoning — that $\triangle PVY$ and $\triangle PYW$ have the same height and bases in ratio $4:3$ — is correct. (A detail: $PY$ is not necessarily the height of these triangles, since it is not necessarily perpendicular to $VW$. But the perpendicular distance from $P$ to $VW$ is the same for both triangles anyway.)
So at this point you have the following areas known and unknown:
Here is a small further step:
The same idea also applies to $\triangle UVY$ and $\triangle UYW$: they have the same height and bases in ratio $4:3$, so their areas are also in ratio $4:3$. We don't know what their areas are, but this does tell us that $$ a+b+40 = \tfrac43 (c+35+30) $$
It would be nice to apply this same idea to some more triangles, to get some more algebraic relations among $a,b,c$, hoping that eventually we'll have enough relations to actually solve for them (or at least for $b$, which is what we were asked for). For example, if we knew the ratio $UX:XV$, then we could draw conclusions about the ratio of the areas $a$ and $b$, and about the ratio of the areas $a+40+30$ and $b+c+35$. Of course, we don't know the ratio $UX:XV$. Can we do something anyway?
Project the moving triangle on the $yz$ plane. This projection preserves areas.
We are now faced with a 2D issue with a moving line. The base triangle has area 12.
The ratio of the area of the moving triangle and the base triangle is equal to the "shrinking" factor
$$f(x)=\left(\dfrac{8-x}{8}\right)^2$$
(for example, if $x=4$, the surface is shortened by a factor $f(4)=1/4$).
It remains to compute the volume of the wedge (or tetrahedron) as the integral of the area of its sections:
$$\int_0^8 12 f(x) dx = 32.$$
Best Answer
The question in this form does not make much sense. You always minimize or maximize something under some conditions.
So, for example, you can minimize or maximize the area of triangles where the three points lie on the unit circle.
What can go wrong:
It can happen that the area of the triangles are arbitrarily large, so that there is no maximal area, but in that case, we can just say so (that the area is arbitrarily large).
It can happen that the area of all triangles under a given condition is smaller than 2, can be chosen arbitrarily close to 2, but never 2. In that case, one calls 2 the supremum instead of the maximum. (The same thing can happen with the minimum.) Again, this can be explicitly stated and proved.
It can happen that the maximum exists, but there is no good explicit way of expressing it. In that case, there is not much you can do, not every number has a good explicit representation.