11.2 Homework: direct and inverse variation

For the following set of data…

• 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.

For the following set of data…

• 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.

For the following set of data…

• 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.

In #2 above, as you (hopefully) saw, the relationship is neither direct nor inverse. However, the relationship can be expressed this way: $y$ is directly proportional to $x^2$ .

• A

  Write the function that indicates this relationship. What is $k$ ?
  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 $x^2$ ?

• C

  What happens to $y$ when you double $x$ ?

• D

  What happens to $y$ when you triple $x$ ?

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.

• 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 $y$ if you double $\sqrt{x}$ ? What happens to $y$ if you multiply $y$ 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?

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.

Joint variation

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 of the two independent variables, $N$ and $C$ .

• D

  Write the general function that states “ $z$ varies jointly as both $x$ and $y$ .” Your function will have an unknown $k$ in it, a constant of variation.