Difference between revisions of "2024 AMC 8 Problems/Problem 24"
Pi is 3.14 (talk | contribs) |
(→Problem) |
||
(34 intermediate revisions by 19 users not shown) | |||
Line 24: | Line 24: | ||
</asy> | </asy> | ||
− | <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 4\sqrt{2} \qquad \textbf{( | + | <math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4\sqrt{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 5\sqrt{2}</math> |
==Solution 1== | ==Solution 1== | ||
− | Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is <math>12\sqrt{2},</math> which means its area is <math>144.</math> Similarly, the area of the second largest right triangle is <math>64</math> (the side length is <math>8\sqrt{2}</math>), and the area of the overlap | + | Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle as shown. |
+ | <asy> | ||
+ | unitsize(.2cm); | ||
+ | draw((0,0)--(8,8)--(11,5)--(18,12)--(30,0)--cycle,linewidth(1)); | ||
+ | draw((11,5)--(6, 0)--(16, 0)--cycle,linewidth(0.5)); | ||
+ | label("$8\sqrt{2}$",(4,4),NW); | ||
+ | label("$12\sqrt{2}$",(24,6),NE); | ||
+ | draw((8,8)--(7.5,7.5)--(8,7)--(8.5,7.5)--cycle); | ||
+ | draw((18,12)--(17.5,11.5)--(18,11)--(18.5,11.5)--cycle); | ||
+ | draw((11,5)--(11,0),dashed); | ||
+ | label("$h$",(11,2.5),E); | ||
+ | </asy> | ||
+ | The side length of the largest right triangle is <math>12\sqrt{2},</math> which means its area is <math>144.</math> Similarly, the area of the second largest right triangle is <math>64</math> (the side length is <math>8\sqrt{2}</math>), and the area of the overlap is <math>h^2</math> (the side length is <math>h\sqrt{2}</math>). Thus, | ||
<cmath>144+64-h^2=183,</cmath> | <cmath>144+64-h^2=183,</cmath> | ||
− | which means that the answer is <math>\boxed{ | + | which means that the answer is <math>\boxed{\mathbf{(B)}\text{ 5}}.</math> |
~BS2012 | ~BS2012 | ||
+ | |||
+ | ==Solution 2 (Using a Ruler)== | ||
+ | You can measure <math>h</math> with a ruler(rulers are allowed on the AMC 8), and see that <math>h</math> is closest to <math>\boxed{\mathbf{(B)}\text{ 5}}.</math> | ||
+ | |||
+ | ~SID-DARTH-VATER | ||
+ | |||
+ | (Please note: This is not the best way, as diagrams are NOT DRAWN TO SCALE) ~Anonymous ~Continuous_Pi | ||
==Video Solution 1 by Math-X (First understand the problem!!!)== | ==Video Solution 1 by Math-X (First understand the problem!!!)== | ||
− | https:// | + | https://youtu.be/BaE00H2SHQM?si=793ml6R_qT79h-aI&t=7651 |
+ | ~Math-X | ||
− | + | ==Video Solution (Quick solution) by Parshwa== | |
+ | https://youtu.be/Ky-NVrOv4ew | ||
− | == | + | ==Video Solution by Power Solve (easy to understand)== |
− | https://www.youtube.com/watch?v= | + | https://www.youtube.com/watch?v=PlMGNmWIkBg |
==Video Solution by OmegaLearn.org== | ==Video Solution by OmegaLearn.org== | ||
https://youtu.be/KL_CGQrkXXo | https://youtu.be/KL_CGQrkXXo | ||
+ | |||
+ | ==Video Solution by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=RiSt6_WLfrM | ||
+ | |||
+ | == Video Solution by NiuniuMaths (Easy to understand!) == | ||
+ | https://www.youtube.com/watch?v=8353ngt05N0 | ||
+ | |||
+ | ~NiuniuMaths | ||
+ | |||
+ | == Video Solution by CosineMethod [🔥Fast and Easy🔥]== | ||
+ | |||
+ | https://www.youtube.com/watch?v=H5Pq8mf-OVk | ||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/ktzijuZtDas&t=3126 | ||
+ | ==2 minute solve (fast and elegant) by MegaMath== | ||
+ | https://www.youtube.com/watch?v=hUh0hux3xuU | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2024|num-b=23|num-a=25}} | ||
+ | {{MAA Notice}} |
Latest revision as of 21:14, 24 August 2024
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2 (Using a Ruler)
- 4 Video Solution 1 by Math-X (First understand the problem!!!)
- 5 Video Solution (Quick solution) by Parshwa
- 6 Video Solution by Power Solve (easy to understand)
- 7 Video Solution by OmegaLearn.org
- 8 Video Solution by SpreadTheMathLove
- 9 Video Solution by NiuniuMaths (Easy to understand!)
- 10 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 11 Video Solution by Interstigation
- 12 2 minute solve (fast and elegant) by MegaMath
- 13 See Also
Problem
Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is feet high while the other peak is
feet high. Each peak forms a
angle, and the straight sides form a
angle with the ground. The artwork has an area of
square feet. The sides of the mountain meet at an intersection point near the center of the artwork,
feet above the ground. What is the value of
Solution 1
Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle as shown.
The side length of the largest right triangle is
which means its area is
Similarly, the area of the second largest right triangle is
(the side length is
), and the area of the overlap is
(the side length is
). Thus,
which means that the answer is
~BS2012
Solution 2 (Using a Ruler)
You can measure with a ruler(rulers are allowed on the AMC 8), and see that
is closest to
~SID-DARTH-VATER
(Please note: This is not the best way, as diagrams are NOT DRAWN TO SCALE) ~Anonymous ~Continuous_Pi
Video Solution 1 by Math-X (First understand the problem!!!)
https://youtu.be/BaE00H2SHQM?si=793ml6R_qT79h-aI&t=7651 ~Math-X
Video Solution (Quick solution) by Parshwa
Video Solution by Power Solve (easy to understand)
https://www.youtube.com/watch?v=PlMGNmWIkBg
Video Solution by OmegaLearn.org
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=RiSt6_WLfrM
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=8353ngt05N0
~NiuniuMaths
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=H5Pq8mf-OVk
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=3126
2 minute solve (fast and elegant) by MegaMath
https://www.youtube.com/watch?v=hUh0hux3xuU
See Also
2024 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.