Re: need help with homework

From: Scena <metarell...@xxxxxxxxx>
Bryana wasn't too sure about the "circuiting" concepts that you were
trying to explain

Think of two homes with separate connections to the electric pole.
But one of the lines goes down, so one house is without power.
So the home with power is cross-circuited to the one without power,
and now both homes are getting power from a single power-line connection.

That's just a metaphor, to explain my jargon. Now to the actual
problem. I like to draw data-flow diagrams for each function,
showing input and output.

We have several totally separate circuits for each calculation involved:

<http://www.webmath.com/geo_cylinder.html>
* The diameter, or the distance across a cylinder that passes
through the center of the cylinder is 2R (twice the radius).

+---------------\
radius ---->| multiply by 2 >----> diameter
+---------------/

/------------+
radius <------< divide by 2 |<----- diameter
\------------+

<http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=2661&langcode=en&expand=0>
If a (right circular) cylinder has height h and base radius r, it has
volume {pi}r^2 h .
A cylinder (of any kind) which has base area A and height h measured
perpendicular to the base, has volume Ah.

+-------\ +-----------------\
radius -->| square >-->| multiply by {pi} >--> area of base
+-------/ +-----------------/

+--------\
area of base -------->| \
| multiply >----> volume of cylinder
height of cylinder -->| /
+--------/

The height of a right cylinder equals its [corrected spelling] diameter.

diameter <=======> height of cylinder

Write the volume of the cylinder
as a function of its [corrected spelling] radius.

So we need something that goes like this:
radius ---> ......... ---> volume of cylinder
So let's hook up all the individual parts into a circult like that
(boxes abbreviated so they'll fit, and to reduce work typing all this):

+-\
radius->[*2]->diameter->heightOfCylinder->| \
| * >-->volumeOfCylinder
radius---->[square]->[*{pi}]->areaOfBase->| /
+-/
So now we have all the math, expressed as function of two
variables, except both inputs are the same. Now we cross-circuit
from one input to the other to get a function of just one variable:
+-\
radius->[*2]->diameter->heightOfCylinder->| \
\ | * >-->volumeOfCylinder
\-->[square]->[*{pi}]->areaOfBase->| /
+-/
Two calculations (houses) feeding off the same input (power line).

Then you can convert that whole diagram to a single math expression.

OT: If you're programming this in FORTH, you must explicitly DUP
the radius so that you have two copies of it to use in the two
separate calculation chains. If you're programming this in PERL,
you make two separate mentions of $1 meaning first parameter.
In most other languages, including regular mathematics, you just
give the radius a name and mention that same name in two different
places. (It's interesting how various functional programming
languages deal with re-using the same value more than once.)

However, she does understand that the "problem"
given by the teacher was to be answered with a hypothetical equation,
rather than with concrete numbers, which is what she learned to do in
geometry.

Yes, abstracting the algebraic (functional) relationships
(formulas) out of geometric knowledge, and combining them to make
new more complicated algebraic relationships (formulas), is a
useful thing to understand, and it's good that your daughter
understood that already.

I thought you just wanted me to go over those questions with
Bryana, not type them out here.

Yeah, I was asking leading questions to see how she responds, so I
can better tailor my help toward her existing knowledge (test and
skip) and missing knowledge (help her more in those areas).
I think you posted one more followup later, so maybe the responses
are there.

Yeah, I did see that. Some idiot going to USC wanted a solution
manual. What's the point in that? Why even bother going to school if
you don't want to learn. A degree is just a piece of paper.

<flame>Yeah, but you'd be amazed how may people lie their ways into
jobs while honest people like me remain unemployed. I was actually
advised at one time that I should stretch my resume to include
experience I don't have but believe I could learn on-the-job if
needed, to better qualify for jobs. I declined the advice, and I'm
still unemployed, while the liars have good jobs, and their
employers are stuck with them.</flame>

Bryana feels that she is a good teacher but moves a little too fast.

In theory, with extra help at home, or after school, she should be
able to keep up with everything the teacher is trying to teach,
unless her total energy level is exceeded. She *does* need sleep too!

Your (parent's) answers to my questions to you separately later ...
.

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