Lesson 6 Towers and Cylinders Solidify Understanding

Learning Focus

Analyze contexts to identify the key features of an inverse variation.

What do I look for to claim that a relationship is an inverse variation in a table, a graph, or an equation?

Open Up the Math: Launch, Explore, Discuss

Mateo’s little sister, Maria, is building towers out of $1 -inch$ cubes. To vary the height of the towers, she makes the base bigger or smaller in area. She has $36$ cubes and wonders how many towers she can build. She is trying to answer this question experimentally, but Mateo thinks he can help her out by listing all possible combinations of tower height as a function of the area of the base.

1.

List all possible combinations of tower height and area of the base that Maria can make with her $36$ $1 -inch$ cubes, then sketch a graph of this relationship.

Possible tower combinations:

Maria’s tower problem reminds Mateo of some of the work he has done in chemistry class measuring liquids with cylindrical beakers. For small amounts of liquid, he has noticed that narrower cylinders work better, since the markings on the side of the beaker are farther apart than they are on fatter beakers, allowing for more precise measurements. Mateo recalls a recent experiment in which he needed to measure out $72 {cm}^{3}$ of liquid and how he had tried wider beakers trying to get a very precise measurement for his experiment.

2.

How would Mateo’s graph of the relationship between height of the liquid and area of the base of the cylinder compare to Maria’s graph representing her towers?

Mateo would like to examine the relationship between the height of liquid in a cylinder and the radius of the base. He wonders if this is also an inverse variation.

3.

Complete this table, graph, and equation for Mateo’s relationship between the height of liquid in a cylinder and the radius of the base for $72 {cm}^{3}$ of liquid. (Recall: The volume formula for a cylinder is $V = π r^{2} h$ . Hint: What form of this equation would make the work of filling out the table more efficient?)

Table:

$r$ (cm)	$h$ (cm)
$1$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$9$
$10$
$11$
$12$

Equation:

Graph:

4.

Is the relationship given by the representations in problem 3 an inverse variation? What is similar and what is different between the way the height of the liquid varies with relationship to the independent variable, area, or radius, as described in problems 2 and 3?

5.

Modify the table you created in problem 3 to be a relationship between the quantities $height$ of liquid in a cylinder and the ${radius}^{2}$ of the base of the cylinder for $72 {cm}^{3}$ of liquid. Is this new relationship an inverse relationship? Why or why not?

$r^{2}$ (cm)	$h$ (cm)
$1$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$9$
$10$
$11$
$12$

Ready for More?

Why is it not sufficient evidence to say that a graph represents an inverse variation by observing characteristic trends, such as noting that as the input quantity gets large, the output quantity approaches $0$ , and as the input quantity approaches $0$ , the output quantity approaches infinity? Write a convincing argument explaining why this is not sufficient to make a claim that the relationship between the input and output quantities is an inverse variation.

Takeaways

The volume formulas for the volume of a right rectangular prism, $V = l \cdot w \cdot h$ , and for a right cylinder, $V = π r^{2} h$ , can both be rewritten in the form , where

This form reveals

To find an inverse or direct variation relationship,

Lesson Summary

In this lesson, we continued to explore inverse variation functions in geometric contexts and found that we may need to strategically choose the quantities related by a function in order to reveal a potential inverse variation relationship. We also learned to be careful when examining the shape of a graph as potentially displaying an inverse function relationship.

Retrieval

1.

What does it mean to have a $50$ / $50$ chance?

Probability is the measure of how likely an event will occur. Probabilities are written as fractions or decimals from $0$ to $1$ , or as percentages from $0 %$ to $100 %$ .

Write not possible, unlikely, as likely as not, likely, or certain to describe each event. The following scale might help you think about your answers.

a.

You were once $10$ years old.

b.

You have taken four tests in math this term and have earned scores of $99 %$ , $94 %$ , $100 %$ , and $97 %$ . You have done your homework and studied. You will earn a score above $95 %$ on today’s test.

c.

Next week will have three Mondays.

2.

Solve for $x$ . Show your steps and check your solution.

$\sqrt{x^{2} + 10 x + 30} = 3$