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Variation Equations (page 1 of 3) Variation problems aren't hard once you get the hang of the lingo. The only real difficulty is learning the somewhat specialized vocabulary and the techniques for this classification of problems.Variation problems involve fairly simple relationships
or formulas, involving one variable being equal to one term. That term might be linear (something
with just an "x"),
quadratic (something in "x2"),
more than one variable (such as "r2h"),
a square root (something like " An example of a variation equation would be the
formula for the area of the circle: On the other hand, "inverse variation" means that the variable is underneath, in the bottom of a fraction. Suppose, for instance, that you inherit a money market account containing $100,000, and you wonder how much money your rich uncle initially invested eight years ago. Depending on the average interest rate "r", the formula you would use would be:
...where P
is the principal your uncle invested. (This formula is a variant of the
compound-interest formula, by the way.)
In the language of variation, this formula reads as "the principal P
varies inversely with The other case of variation is "jointly". "Joint
variation" means "directly, but with two or more variables". An example would be
the formula for the area of a triangle with base "b"
and height "h":
To review:
Be careful with those middle two. Almost always, when you translate word problems from English into math, "and" means "plus" or "added to". But in joint variation, "and" just means "both of these are together on the same side of the fraction" (usually on top), and you multiply. If you are supposed to add two variables, they'll use the format in that third bulleted example above, or they'll say "varies as the sum of x and y. Translating variation problems isn't so bad, once you get the hang of it. But then they want you to move on to setting up and solving word problems. These generally fall into two categories: the ones where they want you to find the value of "k", and the ones where they want you to find some other value, but only after you've found "k" first. Here are some examples:
Since this is direct variation, the formula is "y = kx2". The reason they've given me the data point (x, y) = (2, 8) is that I have to be able to find the value of "k". So I'll plug in the information they've given me, and solve for k: y = kx2
Now that I have k, I can rewrite the formula completely: y = 2x2. With this, I can answer the question they actually asked: "Find y when x = 1." y = 2x2
Then the answer is: y = 2Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
Translating the formula from English to math, I get: y = kxz Plugging in the data point they gave me, and solving for the value of k, I get: 5 = k(3)(4)
Now that I have the value of k, I can plug in the new values, and solve for the new value of y: y = ( 5/12)xz
Then the answer is: y = 5/2 Top | 1 | 2 | 3 | Return to Index Next >>
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Copyright © 2000-2009 Elizabeth Stapel | About | Terms of Use |
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"),
or something else. But it is always just the one term in the formula, multiplied by some number,
usually denoted by "k"
if you don't yet know the number's value; this number k
is called "the constant of variation".
. In
the language of variation, "the area A
varies directly with the square of the radius r";
the constant of variation is 
.
This formula is an example of "direct" variation. "Direct variation" means
that, in the one term of the formula, the variable is "on top".
",
with the constant of variation being k
= 100,000.
. In
words, "the area A
varies jointly with b
and h";
the constant of variation is k
= 1/2.
