Difference between revisions of "2003 AMC 10A Problems/Problem 12"

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== Problem ==
 
== Problem ==
A point <math>(x,y)</math> is randomly picked from inside the rectangle with  vertices <math>(0,0)</math>, <math>(4,0)</math>, <math>(4,1)</math>, and <math>(0,1)</math>. What is the probability that <math>x<y</math>?  
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A point <math>(x,y)</math> is randomly picked from inside the rectangle with  vertices <math>(0,0)</math>, <math>(4,0)</math>, <math>(4,1)</math>, and <math>(0,1)</math>. What is the probability that <math>x>y</math>?  
  
 
<math> \mathrm{(A) \ } \frac{1}{8}\qquad \mathrm{(B) \ } \frac{1}{4}\qquad \mathrm{(C) \ } \frac{3}{8}\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \frac{3}{4} </math>
 
<math> \mathrm{(A) \ } \frac{1}{8}\qquad \mathrm{(B) \ } \frac{1}{4}\qquad \mathrm{(C) \ } \frac{3}{8}\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \frac{3}{4} </math>

Revision as of 15:30, 13 August 2009

Problem

A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x>y$?

$\mathrm{(A) \ } \frac{1}{8}\qquad \mathrm{(B) \ } \frac{1}{4}\qquad \mathrm{(C) \ } \frac{3}{8}\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \frac{3}{4}$

Solution

The rectangle has a width of $4$ and a height of $1$.

The area of this rectangle is $4\cdot1=4$.

The line $x=y$ intersects the rectangle at $(0,0)$ and $(1,1)$.

The area which $x>y$ is the right isosceles triangle with side length $1$ that has vertices at $(0,0)$, $(1,1)$, and $(0,1)$.

The area of this triangle is $\frac{1}{2}\cdot1^{2}=\frac{1}{2}$

Therefore, the probability that $x<y$ is $\frac{\frac{1}{2}}{4}=\frac{1}{8} \Rightarrow A$

See Also

2003 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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All AMC 10 Problems and Solutions
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