Graphs of changing areas

7559 /www/nrich/html/content/id/7559/ 1 2011-07-26T09:37:59 <mdoxml version="1.0"> In the problem <a href="/7534">Changing Areas, Changing Perimeters</a>, 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 $y=\frac{10}{x}$. <mdo:image alt="graph y = 10/x" height="415" src="graph.jpg" width="500"></mdo:image> 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 $10$? </mdoxml> <mdoxml version="1.0"> Matthew from QE Boys' School sent us the following clear analysis: By imagining $x$ and $y$ and the length and width of a rectangle, where $y=10/x$, the obvious connection made is to relate the rectangle's dimensions to something constant. By 'reverse engineering' the expression $y=\frac{10}{x}$ into the form $xy=10$, it is immediately obvious that the area of all described rectangles is $10$. The graph displays symmetry through $y=\pm x$, as making the implied substitutions results in: $$\pm x=\frac{10}{\pm y}$$ This is immediately and obviously the same function, as a factor of $\pm1$ is present on both sides, and can be happily and immediately cancelled, and a multiplication by $\frac{y}{x}$ on both sides restores the original form. The symmetry in $y=-x$ has the consequence that solutions also exist in the negative $x$ and $y$ ranges, despite not being applicable to the original problem. As $x$ gets very large, $y$ becomes very small. This can be seen intuitively from the nature of the division function, but also becomes obvious because $xy=10$. Were $x$ to increase, but $y$ remain the same, the relationship above would no longer hold true, because $(x+\delta x)y=10+\delta xy$, and so, for all non-zero $\delta$, the new $xy$ would exceed 10. For a similar reason and demonstrated by a similar expansion, $y$ cannot increase, and so, hence, must decrease for any and all increases in $x$. It is also of note that $y$ must remain positive, as a positive $x$ multiplied by a negative $y$ will be negative (and hence cannot equal positive 10). As 0 is technically the smallest positive number, $y$ must decrease towards 0, but, as nothing can multiply by 0 to create 10, $y$ must asymptotically approach (and hence never actually reach) 0. The class of graph $y=\frac{a}{x}$, where $a$ is any number, is a scaling of the graph $y=\frac{10}{x}$. By observing that all $y$ values are $\frac{a}{10}$ times those for corresponding $x$ values on the original graph AND that all $x$ values are the same scaling of those for corresponding $y$ values on the original graph, it is impossible that any graph of the described class can intercept $y=\frac{10}{x}$ unless $a=10$, and so the scaling is 1-to-1. The line $y=\frac{P}{2}-x$, equivalent to the observation that $P=2(x+y)$, and hence identical to the definition of perimeter, describes all rectangles with perimeter $P$ when plotted. By substituting the expression for $y$ into the equation $y=10/x$, a (disguised) quadratic in $x$ is formed: $$\frac{10}{x}=\frac{P}{2}-x \Rightarrow 2x^2-Px+20=0 $$ In the normal way for quadratics, it is possible to observe that the equation only has solutions when the discriminant is non-negative: $$P^2-4 \times 2 \times 20 \geq 0 \Rightarrow P^2 \geq 160$$ Hence, the graphs only intersect (and therefore solutions only exist) for $P \geq \sqrt{160}$ for the original graph, or for $P \geq 4\sqrt{a}$ when the reciprocal graph is $y=\frac{a}{x}$. As stated above, $P \geq \sqrt{160}$ for the graph $y=\frac{10}{x}$, and, as this describes rectangles with area 10, the smallest value for $P$ must be $\sqrt{160}=4\sqrt{10}$. Well done to Patrick from Woodbridge School, who sent us <a href="/content/id/7559/Patrick.pdf" class="editorial">this solution</a>, and to Lewis from Colchester Royal Grammar School, who sent us <a href="/content/id/7559/Lewis.pdf" class="editorial">this solution</a>. </mdoxml> <mdoxml version="1.0"> <h3>Why do this problem?</h3> 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. <h3>Possible approach</h3> Display the graph or hand out <a href="/content/id/7559/ChangingAreas1.pdf">this worksheet</a>. "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?" Students should quickly establish that all the rectangles would have the same area. Now display the questions from the problem, or hand out <a href="/content/id/7559/ChangingAreas2.pdf">this sheet</a>. 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. <h3>Key questions</h3> 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 $y=\frac{10}{x}$? <h3>Possible extension</h3> Students could be invited to consider representations in three dimensions of cuboids with equal volume. <h3>Possible support</h3> Set students the stage 3 problem <a href="/6398">Can They Be Equal?</a> as a warm-up before beginning this task. </mdoxml> <mdoxml version="1.0"> How does the rectangle with length $x$ and width $y$ relate to the rectangle with width $x$ and length $y$? </mdoxml> 5 3 0 0 0 0 1 Graphs of changing areas Use graphs to gain insights into an area and perimeter problem, or use your knowledge of area and perimeter of rectangles to gain insight into the graphs. Sequences, Functions and Graphs Rational functions Algebra Simultaneous equations Algebra Solving equations graphically