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Difference between revisions of "2002 AMC 12B Problems/Problem 25"

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(→‎Solution 1)
 
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The area of <math>R</math> is closest to  
 
The area of <math>R</math> is closest to  
  
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<math>\mathrm{(A)}\ 21
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<math>\textbf{(A) } 21
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\qquad\mathrm{(B)}\ 22
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\qquad\textbf{(B)}\ 22
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\qquad\mathrm{(C)}\ 23
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\qquad\textbf{(C)}\ 23
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\qquad\mathrm{(D)}\ 24
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\qquad\textbf{(D)}\ 24
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\qquad\mathrm{(E)}\ 25</math>
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\qquad\textbf{(E)}\ 25</math>
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== Solution ==
+
 
 +
== Solution 1==
 
The first condition gives us that
 
The first condition gives us that
 
<cmath>x^2 + 6x + 1 + y^2 + 6y + 1 \le 0 \Longrightarrow (x+3)^2 + (y+3)^2 \le 16</cmath>
 
<cmath>x^2 + 6x + 1 + y^2 + 6y + 1 \le 0 \Longrightarrow (x+3)^2 + (y+3)^2 \le 16</cmath>
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Thus either
 
Thus either
  
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<cmath>x - y \ge 0,\quad x+y+6 \ge 0</cmath>
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<cmath>x - y \ge 0,\quad x+y+6 \le 0</cmath>
  
 
or
 
or
  
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<cmath>x - y \le 0,\quad x+y+6 \le 0</cmath>
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<cmath>x - y \le 0,\quad x+y+6 \ge 0</cmath>
 +
 
 +
[[Image:2002_12B_AMC-25.png|center]]
 +
 
 +
Each of those lines passes through <math>(-3,-3)</math> and has slope <math>\pm 1</math>, as shown above. Therefore, the area of <math>R</math> is half of the area of the circle, which is <math>\frac{1}{2} (\pi \cdot 4^2) = 8\pi \approx \boxed{\textbf{(E) }25}</math>.
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~SHEN KISLAY KAI
  
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[[Image:2002_12B_AMC-25.png]]
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== Solution 2==
 +
Similar to Solution 1, we proceed to get the area of the circle satisfying <math>f(x)+f(y) \le 0</math>, or <math>16 \pi</math>.
  
−
Each of those lines passes through <math>(-3,-3)</math> and has slope <math>\pm 1</math>, as shown above. Therefore, the area of <math>R</math> is half of the area of the circle, which is <math>\frac{1}{2} (\pi \cdot 4^2) = 8\pi \approx 25 \Rightarrow \mathrm{(E)}</math>.
+
Since <math>f(x)-f(y) \le 0 \implies f(x) \le f(y)</math>, we have that by symmetry, if <math>(x,y)</math> is in <math>R</math>, then <math>(y,x)</math> is not, and vice versa. Therefore, the shaded part of the circle above the line <math>y=x</math> has the same area as the unshaded part below <math>y=x</math>, and the unshaded part above <math>y=x</math> has the same area as the shaded part below <math>y=x</math>. This means that exactly half the circle is shaded, allowing us to divide by two to get <math>\frac{16 \pi }{2} = 8\pi \approx \boxed{\textbf{(E) }25}</math>. ~samrocksnature + ddot1 +Shen kislay kai
  
 
== See also ==
 
== See also ==
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[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 02:49, 26 September 2024

Problem

Let $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that \[f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0\] The area of $R$ is closest to

$\textbf{(A) } 21 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 23 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 25$

Solution 1

The first condition gives us that \[x^2 + 6x + 1 + y^2 + 6y + 1 \le 0 \Longrightarrow (x+3)^2 + (y+3)^2 \le 16\]

which is a circle centered at $(-3,-3)$ with radius $4$. The second condition gives us that

\[x^2 + 6x + 1 - y^2 - 6y - 1 \le 0 \Longrightarrow (x^2 - y^2) + 6(x-y) \le 0 \Longrightarrow (x-y)(x+y+6) \le 0\]

Thus either

\[x - y \ge 0,\quad x+y+6 \le 0\]

or

\[x - y \le 0,\quad x+y+6 \ge 0\]

2002 12B AMC-25.png

Each of those lines passes through $(-3,-3)$ and has slope $\pm 1$, as shown above. Therefore, the area of $R$ is half of the area of the circle, which is $\frac{1}{2} (\pi \cdot 4^2) = 8\pi \approx \boxed{\textbf{(E) }25}$. ~SHEN KISLAY KAI

Solution 2

Similar to Solution 1, we proceed to get the area of the circle satisfying $f(x)+f(y) \le 0$, or $16 \pi$.

Since $f(x)-f(y) \le 0 \implies f(x) \le f(y)$, we have that by symmetry, if $(x,y)$ is in $R$, then $(y,x)$ is not, and vice versa. Therefore, the shaded part of the circle above the line $y=x$ has the same area as the unshaded part below $y=x$, and the unshaded part above $y=x$ has the same area as the shaded part below $y=x$. This means that exactly half the circle is shaded, allowing us to divide by two to get $\frac{16 \pi }{2} = 8\pi \approx \boxed{\textbf{(E) }25}$. ~samrocksnature + ddot1 +Shen kislay kai

See also

2002 AMC 12B (Problems • Answer Key • Resources)
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