OpenStax-CNX module: m19225
1
Modeling Data with Functions
Homework Homework: Direct and
∗
Inverse Variation
Kenny M. Felder
This work is produced by OpenStax-CNX and licensed under the
†
Creative Commons Attribution License 2.0
Abstract
This module provides practice problems related to direct and inverse variation.
note: For #13, please note that these numbers are meant to simulate real world datathat is to
say, they are not necessarily
exact!
If it is darn close to direct or inverse variation, that's good
enough.
Exercise 1
For the following set of data. . .
x
y
3
5
6
11
21
34
Table 1
a. Does it represent direct variation, inverse variation, or neither?
b. If it is direct or inverse, what is the constant of variation?
c. If x = 30, what would y be?
d. Sketch a quick graph of this relationship.
Exercise 2
For the following set of data. . .
∗ Version
1.1: Jan 7, 2009 3:48 pm -0600
† http://creativecommons.org/licenses/by/2.0/
http://cnx.org/content/m19225/1.1/
OpenStax-CNX module: m19225
2
x
y
3
18
4
32
10
200
Table 2
a. Does it represent direct variation, inverse variation, or neither?
b. If it is direct or inverse, what is the constant of variation?
c. If x = 30, what would y be?
d. Sketch a quick graph of this relationship.
Exercise 3
For the following set of data. . .
x
y
3
20
6
10
21
3
Table 3
a. Does it represent direct variation, inverse variation, or neither?
b. If it is direct or inverse, what is the constant of variation?
c. If x = 30, what would y be?
d. Sketch a quick graph of this relationship.
Exercise 4
In #2 above, as you (hopefully) saw, the relationship is neither direct nor inverse. However, the
relationship can be expressed this way:
a.
y
is directly proportional to x2 .
Write the function that indicates this relationship. What is
k?
note: Once you have written the relationship, you can use it to generate more points that
may be helpful in answering part (b)
−
(c).
b. What happens to y when you double x ?
c. What happens to y when you double x?
d. What happens to y when you triplex?
2
Exercise 5
In June, 2007, Poland argued for a change to the voting system in the European Union Council
The Polish suggestion: each member's voting strength should be directly
proportional to the square root of his country's population. This idea, traditionally known
of Ministers.
as Pensore's Rule, is almost sacred among people versed in the game theory of voting according
to one economist.
I swear I am not making this up.
Also in the category of things I am not making up, the following table of European Populations
comes from Wikipedia.
http://cnx.org/content/m19225/1.1/
OpenStax-CNX module: m19225
3
Germany
83,251,851
Italy
59,715,625
Poland
38,625,478
Luxembourg
448,569
Table 4
a.
b.
Write an equation that represents Pensore's Rule. Be sure to clearly label your variables.
Suppose that Pensore's Rule was followed, and suppose that Poland voting strength was
exactly 100 (which I did actually make up, but of course it doesn't matter). What would
c.
d.
the voting strength of Germany, Italy, and Luxembourg be?
Supposing Pensore's Rule is followed. What happens to
to
y
if you multiply
y
y if you double
√
x?
What happens
by 9?
each member's voting strength
should be directly proportional to his country's population. Compared to
Now, suppose a dierent country proposed the rule:
Pensore's Rule, how would that change things? Would it make things better for smaller
countries, better for larger countries, or would it not make a dierence at all?
Exercise 6
Write a real world word problem involving an inverse relationship, on the topic of
Identify the constant of variation.
movies.
Write the function that shows how the dependent variable
depends inversely upon the independent variable.
Create a specic numerical question, and use
your function to answer that question.
Exercise 7
Joint Variation
The term Joint Variation is used to indicate that
variables.
one variable varies directly as two dierent
This is illustrated in the following example.
Al is working as a waiter. When a group of people sit down at a table, he calculates his expected
tip (T ) as follows: multiply the
number of people (N ), times the average meal cost (C ), times
0.15 (for a 15% tip).
a. If the number of people at the table doubles, does Al's expected tip double?
b. If the average cost per meal doubles, does Al's expected tip double?
c. Write the function that expresses the dependent variable, T , as a function
d.
independent variables,
N
and
Write the general function that states z varies jointly as both
will have an unknown
http://cnx.org/content/m19225/1.1/
k
of the
two
C.
in it, a constant of variation.
x
and
y .
Your function