Difference between revisions of "2005 AMC 12B Problems/Problem 7"
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== Solution == | == Solution == | ||
| + | If we get rid of the absolute values, we are left with the following 4 equations (using the logic that if <math>|a|=b</math>, then <math>a</math> is either <math>b</math> or <math>-b</math>): | ||
| + | <cmath><math>\begin{align*} 3x+4y=12 \\ -3x+4y=12 \\ 3x-4y=12 \\ -3x-4y=12 \end{align*}</math></cmath> | ||
| + | |||
| + | We can then put these equations in slope-intercept form in order to graph them. | ||
| + | |||
| + | <cmath><math>\begin{align*} 3x+4y=12 \,\implies\, y=-\dfrac{3}{4}x+3\\ -3x+4y=12\,\implies\, y=\dfrac{3}{4}x+3\\ 3x-4y=12\,\implies\, y=\dfrac{3}{4}x-3\\ -3x-4y=12\,\implies\, y=-\dfrac{3}{4}x-3\end{align*}</math></cmath> | ||
| + | |||
| + | Now you can graph the lines to find the shape of the graph: | ||
| + | |||
| + | <asy> | ||
| + | Label f; | ||
| + | f.p=fontsize(6); | ||
| + | xaxis(-8,8,Ticks(f, 4.0)); | ||
| + | yaxis(-6,6,Ticks(f, 3.0)); | ||
| + | fill((0,-3)--(4,0)--(0,3)--(-4,0)--cycle,grey); | ||
| + | draw((-4,-6)--(8,3), Arrows(4)); | ||
| + | draw((4,-6)--(-8,3), Arrows(4)); | ||
| + | draw((-4,6)--(8,-3), Arrows(4)); | ||
| + | draw((4,6)--(-8,-3), Arrows(4));</asy> | ||
| + | |||
| + | We can easily see that it is a rhombus with diagonals of <math>6</math> and <math>8</math>. The area is <math>\dfrac{1}{2}\times 6\times8</math>, or <math>\boxed{\mathrm{(D)}\ 24}</math> | ||
== See also == | == See also == | ||
| − | + | {{AMC12 box|year=2005|ab=B|num-b=6|num-a=8}} | |
Revision as of 12:17, 4 July 2011
Problem
What is the area enclosed by the graph of 
?
Solution
If we get rid of the absolute values, we are left with the following 4 equations (using the logic that if 
, then 
is either 
or 
):
We can then put these equations in slope-intercept form in order to graph them.
Now you can graph the lines to find the shape of the graph:
![[asy] Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 4.0)); yaxis(-6,6,Ticks(f, 3.0)); fill((0,-3)--(4,0)--(0,3)--(-4,0)--cycle,grey); draw((-4,-6)--(8,3), Arrows(4)); draw((4,-6)--(-8,3), Arrows(4)); draw((-4,6)--(8,-3), Arrows(4)); draw((4,6)--(-8,-3), Arrows(4));[/asy]](http://latex.artofproblemsolving.com/a/1/9/a19def037694543a1771b3897d68d519dd066fcf.png)
We can easily see that it is a rhombus with diagonals of 
and 
. The area is 
, or 
See also
| 2005 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 6 |
Followed by Problem 8 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
