Here are my thoughts regarding your questions.

• If you live in the US, you may consider taking a remedial math course in your local community college.  These don't cost very much, and I think will be more effective than trying to learn the material on your own.  Having a teacher and classmates whom you can talk to can be very helpful.
• I don't know of a product that you can buy to test your mathematical ability, but most universities administer a Math Placement Test to entering students to determine their level of mathematical preparation, and then recommend a suitable course within the mathematical hierarchy to begin one's education.
• You should give some thought as to your reasons for wanting to take a college algebra course.  If it is for the sake of learning some new material, that's fine.  But if you do that as a stepping stone toward a further goal, you should really talk to someone who knows about such matters and seek expert advice before you invest your time, money, and effort toward that goal.

@das1404 Kitonum has already answered your question for pyramids with triangular and square bases. The worksheet pyramid.mw shows how to construct and plot pyramids having n-gons for their bases and the altitudes of whose lateral faces are given.  See if it is of use to you.

Download mw.mw

Do you really need symbolic manipulation capabilities in your research on hyperbolic systems?  To my knowledge, the bulk of computing in that area involves a lot of number crunching but no serious use of symbolic computations.

If that's the case, then the choice between Maple and Mathematica is immaterial.  You may also consider Matlab, or write programs in C or Fortran.  Compilers for C and Fortran are available for free so that choice will cost you $0.

Ultimately the key deciding factor would be which of these languages you feel more comfortable with, and the level of support that you can reasonably expect to receive from its user community.

@das1404 The mathematical definitions a circle and a disk of radius R centered at the origin in the usual Cartesian x-y coordinate system are:

circle    { (x,y) : x² + y² = R² }

disk      { (x,y) : x² + y² ≤ R² }

Maple uses the terms terms "circle" and "disk" according to these standard mathematical definitions.  These should not be confused with the ordinary "street" use of those terms.

The filled versus outline drawings of the plottools objects are controlled through the options style=surface and surface=line.  Try the following to see:

restart;
with(plots): with(plottools):
display(disk([0,0], 1, color=yellow));
display(disk([0,0], 1, color=yellow, style=surface));
display(disk([0,0], 1, color=yellow, style=line));
display([
  disk([0,0], 1, color=red, style=line, thickness=3),
  disk([0,0], 1, color=yellow, style=surface)
]);

Also try the corresponding constructions with circle and rectangle.

PS: I can't tell whether these will have the intended results on Maple 7. They certainly work on the more recent versions of Maple.

The maximum length of 67.66 cm calculated by vv and kitonum is much too small considering that one side of the box is 118 cm long.  That's because the assumption (given in the problem's statement) that the cylinder's axis coincides with the box's diagonal is not good.  If we calculate without that assumption, we arrive at a maximum length of 118.84 cm which makes better sense.  That means that the rolled carpet of length 120 cm is just a bit too long to squeeze in, but perhaps if you give it a little push it may work ;-)

Here is a picture of the box and the longest carpet:

Worksheet: cylinder-in-a-box.mw

Worksheet: mw.mw

Animation extracted fron worksheet mw.mw. The triangle marks the osculating plane. The red dot marks the center of curvature.

I don't have MapleSim but the animation can be produced quite easily in Maple; see the attached worksheet. Here is a sample demonstration where the left end is clamped and the right end is hinged, but you may change the boundary conditions in the worksheet as you wish.

Worksheet: animate-beam.mw

This should be self-explanatory:

Download mw.mw

I find your question quite vague, so here is my attempt to figure out what it is asking.  The calculation that follows is inconclusive.  You need to state more clearly what you have in mind.

Here it is:

Worksheet: mw.mw

You did not provide your equations in the Maple input form, so I retyped them. I don't guarantee that I have made no mistakes. Check!

Download mw.mw

As to your question "Or solve by hand?": I don't think you would want to do that.

General advice: It is important to distinguish between the assignment operator ":=" and the equality operator "=".  The primary purpose of the assignment operator ":=",, is to define labels for expressions to make it easier to refer to them later.  Clearly you don't intend ω as a mere label, rather, you intend ω to enter as a variable in your equations.

So this is what you need to do:

rels1 := diff(phi(t),t) + diff(beta(t),t) = omega(t);
rels2 := diff(rels1, t);
simplify(myvector, {rels1, rels2});

Note how rels1 and rels2 serve as labels for their respective equations.