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Difference between revisions of "2007 iTest Problems/Problem 52"

(Created page with "== Problem == Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both <math>|x|-|y|\le T-500</...")
 
(Solution to Problem 52 — drawing diagram took a lot of work)
 
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== Problem ==
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''The following problem is from the Ultimate Question of the [[2007 iTest]], where solving this problem required the answer of a previous problem.  When the problem is rewritten, the T-value is substituted.''
  
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Let <math>T=TNFTPP</math>. Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both
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==Problem==
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<math>|x|-|y|\le T-500</math> and <math>|y|\le T-500</math>. Find the area of region <math>R</math>.
 
  
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== Solution ==
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Let <math>R</math> be the region consisting of points <math>(x,y)</math> of the Cartesian plane satisfying both
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<math>|x|-|y|\le 16</math> and <math>|y|\le 16</math>. Find the area of region <math>R</math>.
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==Solution==
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From the conditions, <math>-16 \le y \le 16</math> and <math>x \le 16 + |y|</math>.  By analyzing the conditions and testing points, we can graph the inequalities.
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<asy>
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fill((-32,16)--(32,16)--(32,-16)--(-32,-16)--cycle,yellow);
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fill((-32,16)--(-16,0)--(-32,-16)--(-32,-20)--(-4,-20)--(0,-16)--(4,-20)--(32,-20)--(32,-16)--(16,0)--(32,16)--(32,20)--(4,20)--(0,16)--(-4,20)--(-32,20)--cycle,cyan);
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fill((-32,-16)--(32,-16)--(16,0)--(32,16)--(-32,16)--(-16,0)--cycle,green);
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import graph; size(9.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black;
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real xmin=-32.2,xmax=32.2,ymin=-20.2,ymax=20.2;
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pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0);
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/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=2,gy=2;
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for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);
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Label laxis; laxis.p=fontsize(10);
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xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=4.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=4.0,Size=2,NoZero),Arrows(6),above=true);
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clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
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</asy>
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The wanted area is a rectangle with two triangles cut out.  The area of the rectangle is <math>64 \cdot 32 = 2048</math>, and the area of the two triangles is <math>2 \cdot \tfrac12 \cdot 32 \cdot 16 = 512</math>.  That means the area of region <math>R</math> is <math>2048-512 = \boxed{1536}</math>.
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==See Also==
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{{iTest box|year=2007|num-b=51|num-a=53}}
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[[Category:Introductory Algebra Problems]]

Latest revision as of 16:54, 10 July 2018

The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.

Problem

Let $R$ be the region consisting of points $(x,y)$ of the Cartesian plane satisfying both $|x|-|y|\le 16$ and $|y|\le 16$. Find the area of region $R$.

Solution

From the conditions, $-16 \le y \le 16$ and $x \le 16 + |y|$. By analyzing the conditions and testing points, we can graph the inequalities.

[asy] fill((-32,16)--(32,16)--(32,-16)--(-32,-16)--cycle,yellow); fill((-32,16)--(-16,0)--(-32,-16)--(-32,-20)--(-4,-20)--(0,-16)--(4,-20)--(32,-20)--(32,-16)--(16,0)--(32,16)--(32,20)--(4,20)--(0,16)--(-4,20)--(-32,20)--cycle,cyan); fill((-32,-16)--(32,-16)--(16,0)--(32,16)--(-32,16)--(-16,0)--cycle,green); import graph; size(9.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-32.2,xmax=32.2,ymin=-20.2,ymax=20.2; pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0); /*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=2,gy=2; for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs); Label laxis; laxis.p=fontsize(10); xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=4.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=4.0,Size=2,NoZero),Arrows(6),above=true); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

The wanted area is a rectangle with two triangles cut out. The area of the rectangle is $64 \cdot 32 = 2048$, and the area of the two triangles is $2 \cdot \tfrac12 \cdot 32 \cdot 16 = 512$. That means the area of region $R$ is $2048-512 = \boxed{1536}$.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 51 		Followed by:
Problem 53
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4
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