Difference between revisions of "2022 AMC 8 Problems/Problem 24"
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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>\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>\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> | ||
− | 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{\textbf{(C) }192}.</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{\textbf{(C)} ~192}.</math> |
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: |
Revision as of 14:10, 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?
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
After folding polygon on the dotted lines, we obtain the following triangular prism:
~aops-g5-gethsemanea2 ~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.