# Graphs of changing areas

In the problem Changing Areas, Changing Perimeters, you are invited to arrange some rectangles according to their area and perimeter. This problem invites you to consider properties of graphs related to areas and perimeters of rectangles.

The graph below shows the curve $y=\frac{10}{x}$.

Imagine $x$ and $y$ are the length and width of a rectangle.

Each point on the curve represents a rectangle - what property do these rectangles share?

What symmetry does the graph have? How do you know?

What happens to the graph as $x$ gets very large? How do you know?

You could plot graphs of other curves such as $y=\frac{5}{x}$ or $y=\frac{20}{x}$.

How would these graphs relate to the one above? Would the graphs intersect? How do you know?

Rectangles of equal perimeter can be represented graphically by the line $y=\frac{1}{2}P-x$ where $P$ is the perimeter.

Would you expect the line $y=\frac{1}{2}P-x$ to intersect with the curve $y=\frac{10}{x}$ for all values of $P$?

How can you use the graph to find the smallest possible perimeter of a rectangle with an area of $10$?

How does the rectangle with length $x$ and width $y$ relate to the rectangle with width $x$ and length $y$?

The graph shows the relationship between $x$ and $y$ when the area is $10$ square units. Because $y=\frac{10}x, xy=10$. Therefore, the rectangles all have the same area.

The graph has reflectional symmetry in the line $y=x$.

As $x$ becomes very large, the graph will elongate and the $y$ coordinate will near the $x$ axis but will not touch the axis, because the length of a side can never be equal to $0$. This is because as one side increases the other side decreases to maintain a constant area, so in this case an area of $10$ square units.

The graph will never intersect with the graphs $y=\frac{20}x$ and $y=\frac5x$ because the areas of the rectangles have halved or doubled.

Pablo from Kings College Alicante said a little more about the asymptotic behaviour of the graph:

As $x$ gets larger, $y$ tends to $0$

if $x = 10, y = 1$

if $x = 100, y = 0.1$

if $x = 1000, y = 0.01$

etc.

Agathiyan, also from Hymers College, explained why the line $y=\frac{1}{2} P-x$ does not always intersect with the curve $y=\frac{10}x$:

The graph would not intersect for all values of $P$ since the line has a gradient of -1, thus it slopes downwards, and since there is space underneath the curve in the positive $x$ and $y$ quadrant, there are values of the $y$ intercept ($\frac{P}{2}$) which are positive and also allow the line to pass underneath the curve, thus not all values of $P$ intersect. One example of a value that doesn't intersect is if $P=5$.

Pablo showed that a square has the minimum perimeter:

$P = 2x + 2y$

$P = 2x + \frac{20}x$

$P = \frac{2x^2 + 20}{x}$

Differentiating using the quotient rule,

$P' = \frac{(x)(4x) - (2x^2+20)(1)}{x^2}$

$P' = \frac{2x^2-20}{x^2}$

Stationary point when P' = 0

$0 =\frac{2x^2-20}{x^2}$

$0 = 2x^2 - 20$

$20 = 2x^2$

$x^2 = 10$

$x = \sqrt{10}$

$y = \frac{10}{\sqrt{10}} = \sqrt{10}$

In other words, a square has the minimum perimeter of $4\sqrt{10}$. There is no maximum

perimeter because you can make rectangles with sides ${1 000 000, 0.000001}$

which would give a perimeter of over $2 000 000$.

Tim from Gosforth Academy sent us a very clear solution, which you can read as a pdf: Tim's solution.

### Why do this problem?

This problem offers an ideal opportunity to begin thinking about graphs of simple rational functions. Students can begin to make sense of concepts such as symmetry and asymptotes with the security of a concrete example on which to hang their understanding.

### Possible approach

Display the graph or hand out this worksheet.

"What could the graph represent?"

"If I told you that $x$ and $y$ represented the length and width of a family of rectangles, what could you say about all the rectangles?"*All the rectangles have the same area.*

Now display the questions from the problem, or hand out this sheet.

Give students some time to work in pairs to answer the questions. Encourage them to switch between algebraic thinking and reasoning based on their geometrical understanding of the properties of rectangles.

Finally allow some time for students to share their solutions.

### Key questions

How does the rectangle with length $x$ and width $y$ relate to the rectangle with width $x$ and length $y$?

What does it mean when the line $y=\frac{1}{2}P-x$ intersects with the curve $y=\frac{10}{x}$?

### Possible extension

Students could be invited to consider representations in three dimensions of cuboids with equal volume.

### Possible support

Set students the stage 3 problem Can They Be Equal? as a warm-up before beginning this task.