Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2006 AMC 8 Problems/Problem 10"

(Created page with "== Problem == Jorge's teacher asks him to plot all the ordered pairs <math> (w. l) </math> of positive integers for which <math> w </math> is the width and <math> l </math> is t...")
 
m (→‎Solution)
Line 70: Line 70:
 
== Solution ==
 
== Solution ==
  
−
The length of the rectangle will relate invertly to the width, specifically using the theorem <math> l=\frac{12}{w} </math>. The only graph that could represent a inverted relationship is <math> \boxed{\textbf{(E)}} </math>. (The rest are linear graphs that represent dirct relationships)
+
The length of the rectangle will relate invertly to the width, specifically using the theorem <math> l=\frac{12}{w} </math>. The only graph that could represent a inverted relationship is <math> \boxed{\textbf{(E)}} </math>. (The rest are linear graphs that represent direct relationships)

Revision as of 20:20, 6 September 2011

Problem

Jorge's teacher asks him to plot all the ordered pairs $(w. l)$ of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area 12. What should his graph look like?

$\textbf{(A)}$ [asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,12)); dot((2,6)); dot((3,4)); dot((4,3)); dot((6,2)); dot((12,1)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]

$\textbf{(B)}$ [asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,1)); dot((3,3)); dot((5,5)); dot((7,7)); dot((9,9)); dot((11,11)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]

$\textbf{(C)}$ [asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,11)); dot((3,9)); dot((5,7)); dot((7,5)); dot((9,3)); dot((11,1)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]

$\textbf{(D)}$ [asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,6)); dot((3,6)); dot((5,6)); dot((7,6)); dot((9,6)); dot((11,6)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]

$\textbf{(E)}$ [asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((6,1)); dot((6,3)); dot((6,5)); dot((6,7)); dot((6,9)); dot((6,11)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]

Solution

The length of the rectangle will relate invertly to the width, specifically using the theorem $l=\frac{12}{w}$. The only graph that could represent a inverted relationship is $\boxed{\textbf{(E)}}$. (The rest are linear graphs that represent direct relationships)

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_AMC_8_Problems/Problem_10&oldid=42180"