Difference between revisions of "2023 AMC 10B Problems/Problem 24"
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+ | ==Problem== | ||
What is the perimeter of the boundary of the region consisting of all points which can be expressed as <math>(2u-3w, v+4w)</math> with <math>0\le u\le1</math>, <math>0\le v\le1,</math> and <math>0\le w\le1</math>? | What is the perimeter of the boundary of the region consisting of all points which can be expressed as <math>(2u-3w, v+4w)</math> with <math>0\le u\le1</math>, <math>0\le v\le1,</math> and <math>0\le w\le1</math>? | ||
+ | <math>\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16</math> | ||
==Solution 1== | ==Solution 1== | ||
− | Notice that | + | <asy> |
+ | import geometry; | ||
+ | pair A = (-3, 4); | ||
+ | pair B = (-3, 5); | ||
+ | pair C = (-1, 4); | ||
+ | pair D = (-1, 5); | ||
+ | |||
+ | |||
+ | pair AA = (0, 0); | ||
+ | pair BB = (0, 1); | ||
+ | pair CC = (2, 0); | ||
+ | pair DD = (2, 1); | ||
+ | |||
+ | |||
+ | |||
+ | //draw(A--B--D--C--cycle); | ||
+ | |||
+ | |||
+ | |||
+ | draw(A--B); | ||
+ | label("1",midpoint(A--B),W); | ||
+ | label("2",midpoint(D--B),N); | ||
+ | draw(A--C,dashed); | ||
+ | draw(B--D); | ||
+ | draw(C--D, dashed); | ||
+ | |||
+ | draw(A--AA); | ||
+ | label("5",midpoint(A--AA),W); | ||
+ | draw(B--BB,dashed); | ||
+ | draw(C--CC,dashed); | ||
+ | draw(D--DD); | ||
+ | label("5",midpoint(D--DD),E); | ||
+ | label("1",midpoint(CC--DD),E); | ||
+ | label("2",midpoint(AA--CC),S); | ||
+ | |||
+ | // Dotted vertices | ||
+ | dot(A); dot(B); dot(C); dot(D); | ||
+ | |||
+ | |||
+ | |||
+ | dot(AA); dot(BB); dot(CC); dot(DD); | ||
+ | |||
+ | draw(AA--BB,dashed); | ||
+ | draw(AA--CC); | ||
+ | draw(BB--DD,dashed); | ||
+ | draw(CC--DD); | ||
+ | |||
+ | label("(0,0)",AA,W); | ||
+ | label("(-3,4)",A,SW); | ||
+ | label("(-1,5)",D,E); | ||
+ | label("(2,1)",DD,NE); | ||
+ | </asy> | ||
+ | Notice that we are given a parametric form of the region, and <math>w</math> is used in both <math>x</math> and <math>y</math>. We first fix <math>u</math> and <math>v</math> to <math>0</math>, and graph <math>(-3w,4w)</math> from <math>0\le w\le1</math>. When <math>w</math> is <math>0</math>, we have the point <math>(0,0)</math>, and when <math>w</math> is <math>1</math>, we have the point <math>(-3,4)</math>. We see that since this is a directly proportional function, we can just connect the dots like this: | ||
<asy> | <asy> | ||
import graph; | import graph; | ||
Label f; | Label f; | ||
− | + | size(5cm); | |
unitsize(0.7cm); | unitsize(0.7cm); | ||
Line 18: | Line 72: | ||
</asy> | </asy> | ||
− | Now, when we vary <math> | + | Now, when we vary <math>2u</math> from <math>0</math> to <math>2</math>, this line is translated to the right <math>2</math> units: |
<asy> | <asy> | ||
Line 25: | Line 79: | ||
unitsize(0.7cm); | unitsize(0.7cm); | ||
− | + | size(5cm); | |
xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
Line 40: | Line 94: | ||
unitsize(0.7cm); | unitsize(0.7cm); | ||
− | + | size(5cm); | |
xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
Line 57: | Line 111: | ||
unitsize(0.7cm); | unitsize(0.7cm); | ||
− | + | size(5cm); | |
xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
Line 65: | Line 119: | ||
filldraw((0,0)--(-3,4)--(-1,4)--(2,0)--cycle, gray); | filldraw((0,0)--(-3,4)--(-1,4)--(2,0)--cycle, gray); | ||
− | filldraw((0,1)--(-3,5)--(- | + | filldraw((0,1)--(-3,5)--(-1,5)--(2,1)--cycle, gray); |
− | + | draw((0,0)--(0,1),black+dashed); | |
− | + | draw((2,0)--(2,1),black+dashed); | |
− | + | draw((-3,4)--(-3,5),black+dashed); | |
− | |||
</asy> | </asy> | ||
Line 78: | Line 131: | ||
unitsize(0.7cm); | unitsize(0.7cm); | ||
− | + | size(5cm); | |
xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
Line 88: | Line 141: | ||
</asy> | </asy> | ||
− | The length of the boundary is simply <math>1+ | + | The length of the boundary is simply <math>1+2+5+1+2+5</math> (<math>5</math> can be obtained by Pythagorean theorem since we have side lengths <math>3</math> and <math>4</math>.). This equals <math>\boxed{\textbf{(E) }16.}</math> |
+ | |||
+ | ~Technodoggo ~ESAOPS | ||
+ | |||
+ | ==Solution 2== | ||
+ | We can find the "boundary points" and work with our intuition to solve the problem. We set each of <math>u, v, w</math> equal to <math>0, 1</math> for a total of <math>8</math> combinations in <math>u, v, w</math>. We now test each one. | ||
+ | |||
+ | Case 1: <math>u = 0, v = 0, w = 0 \implies (0, 0)</math> | ||
+ | |||
+ | Case 2: <math>u = 0, v = 0, w = 1 \implies (-3, 4)</math> | ||
+ | |||
+ | Case 3: <math>u = 0, v = 1, w = 0 \implies (0, 1)</math> | ||
+ | |||
+ | Case 4: <math>u = 0, v = 1, w = 1 \implies (-3, 5)</math> | ||
+ | |||
+ | Case 5: <math>u = 1, v = 0, w = 0 \implies (2, 0)</math> | ||
+ | |||
+ | Case 6: <math>u = 1, v = 0, w = 1 \implies (-1, 4)</math> | ||
+ | |||
+ | Case 7: <math>u = 1, v = 1, w = 0 \implies (2, 1)</math> | ||
+ | |||
+ | Case 8: <math>u = 1, v = 1, w = 1 \implies (-1, 5)</math> | ||
+ | |||
+ | When graphed on a coordinate plane, the points appear as follows. | ||
+ | |||
+ | <asy> | ||
+ | |||
+ | import graph; | ||
+ | import geometry; | ||
+ | Label f; | ||
+ | |||
+ | size(5cm); | ||
+ | unitsize(0.7cm); | ||
+ | |||
+ | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
+ | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
+ | |||
+ | pair A = (0, 0); | ||
+ | dot (A); | ||
+ | pair B = (-3, 4); | ||
+ | dot (B); | ||
+ | pair C = (0, 1); | ||
+ | dot (C); | ||
+ | pair D = (-3, 5); | ||
+ | dot (D); | ||
+ | pair E = (2, 0); | ||
+ | dot (E); | ||
+ | pair F = (-1, 4); | ||
+ | dot (F); | ||
+ | pair G = (2, 1); | ||
+ | dot (G); | ||
+ | pair H = (-1, 5); | ||
+ | dot (H); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | Notice how there are two distinct rectangles visible in the figure. This leads us to believe that the region tracks the motion of this region as it travels in space. To understand why this is true, we can imagine a fixed <math>w</math> (as it is present in both the <math>x</math> and <math>y</math> coordinates). Then if we hold one of <math>u</math> or <math>v</math> fixed and let the other vary, we get a straight line parallel to the <math>x</math> or <math>y</math> axis respectively. If we let the other vary, we get the other type of straight line. Together, they form a rectangular region. In addition, <math>w</math> serves as a diagonal translation, so if we now let <math>w</math> vary, it traces out the motion of the rectangle. Keeping this in mind, we connect the dots. | ||
+ | |||
+ | <asy> | ||
+ | |||
+ | import graph; | ||
+ | import geometry; | ||
+ | Label f; | ||
+ | |||
+ | size(5cm); | ||
+ | unitsize(0.7cm); | ||
+ | |||
+ | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
+ | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
+ | |||
+ | draw((0,0)--(-3,4)--(-3,5)--(-1,5)--(2,1)--(2,0)--cycle, gray); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | Each of the diagonal sides have length <math>5</math> by the distance formula on <math>(0,0)</math> and <math>(-3,4)</math> (the other diagonal side is congruent), so our total area is <math>2 + 1 + 5 + 2 + 1 + 5 = \boxed{\textbf{(E)}~ 16}</math>. | ||
+ | |||
+ | ~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi] | ||
+ | |||
+ | ==Video Solution 1 by OmegaLearn== | ||
+ | https://youtu.be/KEV3ka5gWYU | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://youtu.be/bqIlsWTOL3k | ||
+ | |||
+ | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
− | + | ==See also== | |
+ | {{AMC10 box|year=2023|ab=B|num-b=23|num-a=25}} | ||
+ | {{MAA Notice}} |
Latest revision as of 12:34, 24 July 2024
Contents
Problem
What is the perimeter of the boundary of the region consisting of all points which can be expressed as with
,
and
?
Solution 1
Notice that we are given a parametric form of the region, and
is used in both
and
. We first fix
and
to
, and graph
from
. When
is
, we have the point
, and when
is
, we have the point
. We see that since this is a directly proportional function, we can just connect the dots like this:
Now, when we vary from
to
, this line is translated to the right
units:
We know that any points in the region between the line (or rather segment) and its translation satisfy and
, so we shade in the region:
We can also shift this quadrilateral one unit up, because of . Thus, this is our figure:
The length of the boundary is simply (
can be obtained by Pythagorean theorem since we have side lengths
and
.). This equals
~Technodoggo ~ESAOPS
Solution 2
We can find the "boundary points" and work with our intuition to solve the problem. We set each of equal to
for a total of
combinations in
. We now test each one.
Case 1:
Case 2:
Case 3:
Case 4:
Case 5:
Case 6:
Case 7:
Case 8:
When graphed on a coordinate plane, the points appear as follows.
Notice how there are two distinct rectangles visible in the figure. This leads us to believe that the region tracks the motion of this region as it travels in space. To understand why this is true, we can imagine a fixed (as it is present in both the
and
coordinates). Then if we hold one of
or
fixed and let the other vary, we get a straight line parallel to the
or
axis respectively. If we let the other vary, we get the other type of straight line. Together, they form a rectangular region. In addition,
serves as a diagonal translation, so if we now let
vary, it traces out the motion of the rectangle. Keeping this in mind, we connect the dots.
Each of the diagonal sides have length by the distance formula on
and
(the other diagonal side is congruent), so our total area is
.
~ cxsmi
Video Solution 1 by OmegaLearn
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.