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Difference between revisions of "2022 AMC 8 Problems/Problem 24"
 
(43 intermediate revisions by 14 users not shown)
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<asy>
 
<asy>
 
usepackage("mathptmx");
 
usepackage("mathptmx");
−
unitsize(1cm);
+
size(275);
−
defaultpen(linewidth(0.7)+fontsize(11));
+
defaultpen(linewidth(0.8));
 
real r = 2, s = 2.5, theta = 14;
 
real r = 2, s = 2.5, theta = 14;
 
pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta);
 
pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta);
Line 19: Line 19:
 
dot("$D$",D,dir(0));
 
dot("$D$",D,dir(0));
 
dot("$E$",E,S);
 
dot("$E$",E,S);
−
dot("$F$",F,1.5*S);
+
dot("$F$",F,1.5*dir(-100));
 
dot("$G$",G,S);
 
dot("$G$",G,S);
 
dot("$H$",H,W);
 
dot("$H$",H,W);
Line 26: Line 26:
 
</asy>
 
</asy>
  
−
<math>\textbf{(A)} ~112\qquad\textbf{(B)} ~128\qquad\textbf{(C)} ~192\qquad\textbf{(D)} ~240\qquad\textbf{(E)} ~288\qquad</math>
+
<math>\textbf{(A)} ~112\qquad\textbf{(B)} ~128\qquad\textbf{(C)} ~192\qquad\textbf{(D)} ~240\qquad\textbf{(E)} ~288</math>
  
 
==Solution==
 
==Solution==
  
−
We try to visualize the prism by folding it in our heads. Then, <math>\overline{AB}</math> goes on <math>\overline{BC},</math> <math>\overline{AH}</math> goes on <math>\overline{CI},</math> and <math>\overline{EF}</math> goes on <math>\overline{FG}.</math> So, <math>\overline{BJ}=\overline{CI}=8</math> and <math>\overline{FG}=\overline{BC}=8.</math> Also, <math>\overline{HJ}</math> becomes an edge parallel to <math>\overline{FG},</math> so that means <math>\overline{HJ}=8.</math>
+
While imagining the folding, <math>\overline{AB}</math> goes on <math>\overline{BC},</math> <math>\overline{AH}</math> goes on <math>\overline{CI},</math> and <math>\overline{EF}</math> goes on <math>\overline{FG}.</math> So, <math>BJ=CI=8</math> and <math>FG=BC=8.</math> Also, <math>\overline{HJ}</math> becomes an edge parallel to <math>\overline{FG},</math> so that means <math>HJ=8.</math>
  
−
Since <math>\overline{GH}=14,</math> then <math>\overline{JG}=14-8=6.</math> So, the area of <math>\triangle BJG</math> is <math>\frac{8\cdot6}{2}=24.</math> If we let <math>\triangle BJG</math> be the base, then the height is <math>\overline{FG}=8.</math> So, the volume is <math>24\cdot8=\boxed{192~\textbf{(C)}}.</math>
+
Since <math>GH=14,</math> then <math>JG=14-8=6.</math> So, the area of <math>\triangle BJG</math> is <math>\frac{8\cdot6}{2}=24.</math> If we let <math>\triangle BJG</math> be the base, then the height is <math>FG=8.</math> So, the volume is <math>24\cdot8=\boxed{\textbf{(C)} ~192}.</math>
  
−
Solution by aops-g5-gethsemanea2
+
~aops-g5-gethsemanea2
 +
 
 +
==Remark==
 +
After folding polygon <math>ABCDEFGH</math> on the dotted lines, we obtain the following triangular prism:
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
usepackage("mathptmx");
 +
size(200);
 +
defaultpen(linewidth(0.8));
 +
import graph3;
 +
import solids;
 +
 
 +
currentprojection=orthographic((0.3,-0.3,0.3));
 +
triple J, G, B, A, H, F;
 +
J = (0,0,0);
 +
G = (6,0,0);
 +
B = (0,8,0);
 +
A = (0,8,8);
 +
H = (0,0,8);
 +
F = (6,0,8);
 +
draw(surface(B--J--G--cycle),yellow);
 +
draw(surface(H--A--F--cycle),yellow);
 +
draw(B--J,dashed);
 +
draw(G--J--H--A--B^^A--F--H^^F--G^^B--G);
 +
draw((0.5,0,0)--(0.5,0.5,0)--(0,0.5,0)^^(0.5,0,8)--(0.5,0.5,8)--(0,0.5,8));
 +
dot("$A=C$",A,1.5*E);
 +
dot("$B$",B,1.5*E);
 +
dot("$D=J$",J,1.5*W);
 +
dot("$F$",F,1.5*E);
 +
dot("$H=I$",H,1.5*W);
 +
dot("$E=G$",G,1.5*E);
 +
label("$8$",midpoint(A--H),1.5*NW,red);
 +
label("$6$",midpoint(H--F),1.5*S,red);
 +
label("$8$",midpoint(H--J),1.5*W,red);
 +
</asy>
 +
~MRENTHUSIASM
 +
 
 +
==Video Solution by Math-X (First understand the problem!!!)==
 +
https://youtu.be/oUEa7AjMF2A?si=IKVkcHgWtsS8Fmbs&t=4943
 +
 
 +
~Math-X
 +
 
 +
==Video Solution(🚀Under 2 min🚀 With color-coded folding explanation ✨)==
 +
https://youtu.be/FhQROS_83iw
 +
 
 +
<i>~Education, the Study of Everything</i>
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/MOcX5BFbcwU?t=380
 +
 
 +
~ pi_is_3.14
 +
 
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=2uoBPp4Kxck
 +
 
 +
~Mathematical Dexterity
 +
 
 +
==Video Solution==
 +
https://youtu.be/Ij9pAy6tQSg?t=2432
 +
 
 +
~Interstigation
 +
 
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=tQecx6F1O8k
 +
 
 +
~David
 +
 
 +
==Video Solution==
 +
https://youtu.be/0orAAUaLIO0?t=469
 +
 
 +
~STEMbreezy
 +
 
 +
==Video Solution==
 +
https://youtu.be/YJceP3zGSEU
 +
 
 +
~savannahsolver
 +
 
 +
==Video Solution==
 +
https://youtu.be/_yj11gFIdw4
 +
 
 +
Please like and subscribe!
 +
 
 +
==See Also==
 +
{{AMC8 box|year=2022|num-b=23|num-a=25}}
 +
{{MAA Notice}}

Latest revision as of 14:03, 23 November 2023

Problem

The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism?

[asy] usepackage("mathptmx"); size(275); defaultpen(linewidth(0.8)); real r = 2, s = 2.5, theta = 14; pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta); pair N = (B+G)/2, J = N + s/2 * dir(180+theta); pair E = F + r * dir(- 45 - theta/2), D = I+E-F; pair H = J + r * dir(135 + theta/2), A = B+H-J; draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B)); draw(J--B--G^^C--F--I,linetype ("4 4")); dot("$A$",A,N); dot("$B$",B,1.2*N); dot("$C$",C,N); dot("$D$",D,dir(0)); dot("$E$",E,S); dot("$F$",F,1.5*dir(-100)); dot("$G$",G,S); dot("$H$",H,W); dot("$I$",I,NE); dot("$J$",J,1.5*S); [/asy]

$\textbf{(A)} ~112\qquad\textbf{(B)} ~128\qquad\textbf{(C)} ~192\qquad\textbf{(D)} ~240\qquad\textbf{(E)} ~288$

Solution

While imagining the folding, $\overline{AB}$ goes on $\overline{BC},$ $\overline{AH}$ goes on $\overline{CI},$ and $\overline{EF}$ goes on $\overline{FG}.$ So, $BJ=CI=8$ and $FG=BC=8.$ Also, $\overline{HJ}$ becomes an edge parallel to $\overline{FG},$ so that means $HJ=8.$

Since $GH=14,$ then $JG=14-8=6.$ So, the area of $\triangle BJG$ is $\frac{8\cdot6}{2}=24.$ If we let $\triangle BJG$ be the base, then the height is $FG=8.$ So, the volume is $24\cdot8=\boxed{\textbf{(C)} ~192}.$

~aops-g5-gethsemanea2

Remark

After folding polygon $ABCDEFGH$ on the dotted lines, we obtain the following triangular prism: [asy] /* Made by MRENTHUSIASM */ usepackage("mathptmx"); size(200); defaultpen(linewidth(0.8)); import graph3; import solids; currentprojection=orthographic((0.3,-0.3,0.3)); triple J, G, B, A, H, F; J = (0,0,0); G = (6,0,0); B = (0,8,0); A = (0,8,8); H = (0,0,8); F = (6,0,8); draw(surface(B--J--G--cycle),yellow); draw(surface(H--A--F--cycle),yellow); draw(B--J,dashed); draw(G--J--H--A--B^^A--F--H^^F--G^^B--G); draw((0.5,0,0)--(0.5,0.5,0)--(0,0.5,0)^^(0.5,0,8)--(0.5,0.5,8)--(0,0.5,8)); dot("$A=C$",A,1.5*E); dot("$B$",B,1.5*E); dot("$D=J$",J,1.5*W); dot("$F$",F,1.5*E); dot("$H=I$",H,1.5*W); dot("$E=G$",G,1.5*E); label("$8$",midpoint(A--H),1.5*NW,red); label("$6$",midpoint(H--F),1.5*S,red); label("$8$",midpoint(H--J),1.5*W,red); [/asy] ~MRENTHUSIASM

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/oUEa7AjMF2A?si=IKVkcHgWtsS8Fmbs&t=4943

~Math-X

Video Solution(🚀Under 2 min🚀 With color-coded folding explanation ✨)

https://youtu.be/FhQROS_83iw

~Education, the Study of Everything

Video Solution by OmegaLearn

https://youtu.be/MOcX5BFbcwU?t=380

~ pi_is_3.14

Video Solution

https://www.youtube.com/watch?v=2uoBPp4Kxck

~Mathematical Dexterity

Video Solution

https://youtu.be/Ij9pAy6tQSg?t=2432

~Interstigation

Video Solution

https://www.youtube.com/watch?v=tQecx6F1O8k

~David

Video Solution

https://youtu.be/0orAAUaLIO0?t=469

~STEMbreezy

Video Solution

https://youtu.be/YJceP3zGSEU

~savannahsolver

Video Solution

https://youtu.be/_yj11gFIdw4

Please like and subscribe!

See Also

2022 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

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