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Homework: Direct and Inverse Variation

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides practice problems related to direct and inverse variation.

Note:

For #1–3, please note that these numbers are meant to simulate real world data—that 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…

Table 1
xx yy
3 5
6 11
21 34
  • 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=30x=30, what would yy be?
  • d. Sketch a quick graph of this relationship.

Exercise 2

For the following set of data…

Table 2
xx yy
3 18
4 32
10 200
  • 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=30x=30, what would yy be?
  • d. Sketch a quick graph of this relationship.

Exercise 3

For the following set of data…

Table 3
xx yy
3 20
6 10
21 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=30x=30, what would yy 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: yy is directly proportional to x2x2.

  • a. Write the function that indicates this relationship. What is kk?

    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 yy when you double x2x2?
  • c. What happens to yy when you double xx?
  • d. What happens to yy when you triple xx?

Exercise 5

In June, 2007, Poland argued for a change to the voting system in the European Union Council of Ministers. 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 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.

Table 4
Germany 83,251,851
Italy 59,715,625
Poland 38,625,478
Luxembourg 448,569
  • a. Write an equation that represents Pensore’s Rule. Be sure to clearly label your variables.
  • b. 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 the voting strength of Germany, Italy, and Luxembourg be?
  • c. Supposing Pensore’s Rule is followed. What happens to yy if you double xx? What happens to yy if you multiply yy by 9?
  • d. Now, suppose a different country proposed the rule: each member’s voting strength should be directly proportional to his country’s population. Compared to 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 difference at all?

Exercise 6

Write a “real world” word problem involving an inverse relationship, on the topic of movies. Identify the constant of variation. Write the function that shows how the dependent variable depends inversely upon the independent variable. Create a specific numerical question, and use your function to answer that question.

Exercise 7

Joint Variation

The term “Joint Variation” is used to indicate that one variable varies directly as two different variables. 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 (TT) as follows: multiply the number of people (NN), times the average meal cost (CC), 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, TT, as a function of the two independent variables, NN and CC.
  • d. Write the general function that states “zz varies jointly as both xx and yy.” Your function will have an unknown kk in it, a constant of variation.

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