Sanderson M. Smith

# NOTE TO ALG. II STUDENTS ON SYSTEMS

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Hi ALG. IIers.....

We're doing some very sophisticated problems.

However, they are far from difficult if you read carefully, work neatly, and communicate.I'm going to present problem a sophisticated-sounding problem.The purpose here is not to show you how to do it, but to try to convince you that you can do it if you will read ... and think. As you look at the problem below, and follow the steps, ask yourself what is hard about it. I hope you will conclude that nothing about it is difficult. I'm also going to communicate to you through typing. (Yes, this can be done in math!) Note the proper use of ==> (which can be read "implies").Here's the problem:

A RECTANGULAR PIECE OF CARDBOARD, WHOSE ARE IS 216 SQUARE CENTIMETERS, IS MADE INTO AN OPEN BOX BY CUTTING A 2-CENTIMETER SQUARE FROM EACH CORNER AND TURNING UP THE SIDES. IF THE BOX IS TO HAVE A VOLUME OF 224 CUBIC INCHES, WHAT SIZE CARDBOARD SHOULD YOU START WITH?

 XX side XX side base side XX side XX

Now I'm going to make an executive decision:

Let L = length of cardboard (cm.) and

W = width of cardboard (cm.)

One relation between L and W is

(a): LW = 216.

Now,when we cut out the squares from the corners and form the box, the base of the box has dimensions L-4 and W-4, and the height of the box is 2. (Unit is centimeters).

Another condition on L and W is then

(b): (L-4)(W-4)2 = 224 ==> (L-4)(W-4) = 112.

OK, we now have a system of two equations and two unknowns. The system is

(a): LW = 216

(b): (L-4)(W-4) = 112

We need only solve this system to get the desired answer. The system could be solved algebraically, but I'll set it up for a calculator solution. (And I'll communicate in the process.) My strategy is to solve (a) and (b) for L. This does require a bit of first-year algebra.

(a) ==> L = 216/W

(b) ==> L - 4 = 112/(W-4) ==> L = 112/(W-4) + 4

OK, I am set for a calculator solution. If I now think of L = Y and W = X, I'll get a calculator solution by graphing

Y1 = 216/X

Y2 = 112/(X-4) + 4

Of course, I can waste a lot of time if I don't consider the WINDOW. It should strike you that only positive values for X and Y are meaningful here. (You are clearly wasting window space if you take the STANDARD option.) Reasonable upper limits for X and Y are somewhat of a guess, but make a reasonable guess, take your shot at it, and adjust as necessary.

To be successful in courses like English and history, you must read... and read carefully, and understand what you read. Math is a language, and if you'll learn to apply the same standards of careful reading to it, and apply the same writing standards (use the language properly, etc.), there is nothing difficult about mathematics. It's only difficult if you lack the patience to train yourself to work carefully.

If you can follow what I wrote above, then you are reading the language. Note that I have been careful with my use of = and ==>, and you should be able to tell where everything comes from. If you do read (and understand) what is above, ask yourself what's hard about it. I hope you will conclude there is nothing hard about it. A basic knowledge of geometry and first-year algebra is required, but you have that!

MATH IS POWER.

MAY YOU ALL OBTAIN IT.

-Have a good day.

Doc

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"He who can properly define and divide is to be considered a god."

(Plato, ca 400 B.C.)

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"The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics."

(Johannes Kepler, 1571-1630)

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"Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform."

(Bertrand Russell, 1872-1970)

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"It is not enough to have a good mind. The main thing is to use it well. With me everything turns into mathematics."

(Rene Descartes, 1596-1650)

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"God is a mathematician."

(Karl Friedrich Gauss, 1777-1855)

Note: Gauss, Isaac Newton, and Archimedes are considered by many to be the

three greatest mathematicians of all time.)

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"We are servants rather than masters in mathematics."

(Charles Hermite, 1822-1901)

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