11.1 Homework: inverse variation

An astronaut in space is performing an experiment with three balloons. The balloons are all different sizes but they have the same amount of air in them. As you might expect, the balloons that are very small experience a great deal of air pressure (the air inside pushing out on the balloon); the balloons that are very large, experience very little air pressure. He measures the volumes and pressures, and comes up with the following chart.

• A

  Which is the dependent variable, and which is the independent variable?

• B

  When the volume doubles , what happens to the pressure?

• C

  When the volume triples , what happens to the pressure?

• D

  Based on your answers to parts (a) – (c), what would you expect the pressure to be for a balloon with a volume of 30?

• E

  On the right of the table add a third column that represents the quantity $\mathrm{PV}$ : pressure times volume. Fill in all four values for this quantity. What do you notice about them?

• F

  Plot all four points on the graph paper, and fill in a sketch of what the graph looks like.

• G

  Write the function $P\left(V\right)$ . Make sure that it accurately gets you from the first column to the second in all four instances! (Part (e) is a clue to this.)

• H

  Graph your function $P\left(V\right)$ on your calculator, and copy the graph onto the graph paper. Does it match your graph in part (f)?

The three little pigs have built three houses—made from straw, Lincoln Logs®, and bricks, respectively. Each house is 20' high. The pieces of straw are 1/10" thick; the Lincoln Logs® are 1" thick; the bricks are 4" thick. Let $t$ be the thickness of the building blocks, and let $n$ be the number of such blocks required to build a house 20' high.

• Make a table showing different $t$ values and their corresponding $n$ values.

• A

  Which is the dependent variable, and which is the independent variable?

• B

  When the thickness of the building blocks doubles , what happens to the number required? (*Not sure? Pretend that the pig’s cousin used 8" logs, and his uncle used 16" logs. See what happens to the number required as you go up in this sequence…)

• C

  When the thickness of the building blocks is halved , what happens to the number required?

• D

  On the right of the table add a fourth column that represents the quantity $\mathrm{tn}$ : thickness times number. Fill in all three values for this quantity. What do you notice about them? What do they actually represent, in our problem?

• E

  Plot all three points on the graph paper, and fill in a sketch of what the graph looks like.

• F

  Write the function $n\left(t\right)$ .

• G

  Graph your function $n\left(t\right)$ on your calculator, and copy the graph onto the graph paper. Does it match your graph in part (f)?

The above two scenarios are examples of inverse variation . If a variable $y$ “varies inversely” with $x$ , then it can be written as a function $y=\frac{k}{x}$ , where $k$ is called the constant of variation . So, if $y$ varies inversely with $x$ …

• A

  What happens to $y$ if $x$ doubles?
  You can find and prove the answer from the equation $y=\frac{k}{x}$ .

• B

  What happens to $y$ if $x$ is cut in half?

• C

  What does the graph $y\left(x\right)$ look like? What happens to this graph when $k$ increases? (*You may want to try a few different ones on your calculator to see the effect $k$ has.)

Make up a word problem like #1 and #2 above. Your problem should not involve pressure and volume, or building a house. It should involve two variables that vary inversely with each other. Make up the scenario, define the variables, and then do problems (a) - (h) exactly like my two problems.