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2007 iTest Problems/Problem 52

Revision as of 15:54, 10 July 2018 by Rockmanex3 (talk | contribs) (Solution to Problem 52 — drawing diagram took a lot of work)
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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.

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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