Difference between revisions of "2022 AMC 8 Problems/Problem 24"
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Solution by aops-g5-gethsemanea2 | Solution by aops-g5-gethsemanea2 | ||
| − | == | + | ==Solution 3== |
After folding polygon <math>ABCDEFGH</math> on the dotted lines, we obtain the following triangular prism: | After folding polygon <math>ABCDEFGH</math> on the dotted lines, we obtain the following triangular prism: | ||
| Line 70: | Line 70: | ||
label("$8$",midpoint(H--J),1.5*W,red); | label("$8$",midpoint(H--J),1.5*W,red); | ||
</asy> | </asy> | ||
| + | |||
| + | It is easy to see that the volume of this prism is $\frac {8 \times 6}{2} \times 8 = \boxed {\textbf {(C) }192} | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
Revision as of 08:51, 10 June 2022
Problem
The figure below shows a polygon 
, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that 
and 
. What is the volume of the prism?
![[asy] usepackage("mathptmx"); size(275); defaultpen(linewidth(0.8)); real r = 2, s = 2.5, theta = 14; pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta); pair N = (B+G)/2, J = N + s/2 * dir(180+theta); pair E = F + r * dir(- 45 - theta/2), D = I+E-F; pair H = J + r * dir(135 + theta/2), A = B+H-J; draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B)); draw(J--B--G^^C--F--I,linetype ("4 4")); dot("$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$",J,1.5*S); [/asy]](http://latex.artofproblemsolving.com/8/c/0/8c0183579e57a39614f2c3be1824f286b6bf4b86.png)
Solution
While imagining the folding, 
goes on 
goes on 
and 
goes on 
So, 
and 
Also, 
becomes an edge parallel to 
so that means 
Since 
then 
So, the area of 
is 
If we let be the base, then the height is 
So, the volume is 
Solution by aops-g5-gethsemanea2
Solution 3
After folding polygon on the dotted lines, we obtain the following triangular prism:
![[asy] /* Made by MRENTHUSIASM */ usepackage("mathptmx"); size(200); defaultpen(linewidth(0.8)); import graph3; import solids; currentprojection=orthographic((0.3,-0.3,0.3)); triple J, G, B, A, H, F; J = (0,0,0); G = (6,0,0); B = (0,8,0); A = (0,8,8); H = (0,0,8); F = (6,0,8); draw(surface(B--J--G--cycle),yellow); draw(surface(H--A--F--cycle),yellow); draw(B--J,dashed); draw(G--J--H--A--B^^A--F--H^^F--G^^B--G); draw((0.5,0,0)--(0.5,0.5,0)--(0,0.5,0)^^(0.5,0,8)--(0.5,0.5,8)--(0,0.5,8)); dot("$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]](http://latex.artofproblemsolving.com/3/1/2/312686e55a4760716ba0dafc8635f2217e326d69.png)
It is easy to see that the volume of this prism is $\frac {8 \times 6}{2} \times 8 = \boxed {\textbf {(C) }192} ~MRENTHUSIASM
Video Solution
https://www.youtube.com/watch?v=2uoBPp4Kxck
~Mathematical Dexterity
Video Solution
https://youtu.be/Ij9pAy6tQSg?t=2432
~Interstigation
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
