Difference between revisions of "2023 AMC 8 Problems/Problem 9"
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==Problem== | ==Problem== | ||
Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between <math>4</math> and <math>7</math> meters? | Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between <math>4</math> and <math>7</math> meters? | ||
− | + | <asy> | |
+ | // Diagram by TheMathGuyd. Found cubic, so graph is perfect. | ||
+ | import graph; | ||
+ | size(8cm); | ||
+ | int i; | ||
+ | for(i=1; i<9; i=i+1) | ||
+ | { | ||
+ | draw((-0.2,2i-1)--(16.2,2i-1), mediumgrey); | ||
+ | draw((2i-1,-0.2)--(2i-1,16.2), mediumgrey); | ||
+ | draw((-0.2,2i)--(16.2,2i), grey); | ||
+ | draw((2i,-0.2)--(2i,16.2), grey); | ||
+ | } | ||
+ | Label f; | ||
+ | f.p=fontsize(6); | ||
+ | xaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); | ||
+ | yaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); | ||
+ | real f(real x) | ||
+ | { | ||
+ | return -0.03125 x^(3) + 0.75x^(2) - 5.125 x + 14.5; | ||
+ | } | ||
+ | draw(graph(f,0,15.225),currentpen+1); | ||
+ | real dpt=2; | ||
+ | real ts=0.75; | ||
+ | transform st=scale(ts); | ||
+ | label(rotate(90)*st*"Elevation (meters)",(-dpt,8)); | ||
+ | label(st*"Time (seconds)",(8,-dpt)); | ||
+ | </asy> | ||
<math>\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math> | <math>\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math> | ||
==Solution 1== | ==Solution 1== | ||
− | + | We mark the time intervals in which Malaika's elevation is between <math>4</math> and <math>7</math> meters in red, as shown below: | |
+ | <asy> | ||
+ | // Diagram by TheMathGuyd. Found cubic, so graph is perfect. | ||
+ | import graph; | ||
+ | size(8cm); | ||
+ | int i; | ||
+ | for(i=1; i<9; i=i+1) | ||
+ | { | ||
+ | draw((-0.2,2i-1)--(16.2,2i-1), mediumgrey); | ||
+ | draw((2i-1,-0.2)--(2i-1,16.2), mediumgrey); | ||
+ | draw((-0.2,2i)--(16.2,2i), grey); | ||
+ | draw((2i,-0.2)--(2i,16.2), grey); | ||
+ | } | ||
+ | Label f; | ||
+ | f.p=fontsize(6); | ||
+ | xaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); | ||
+ | yaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); | ||
+ | real f(real x) | ||
+ | { | ||
+ | return -0.03125 x^(3) + 0.75x^(2) - 5.125 x + 14.5; | ||
+ | } | ||
+ | draw(graph(f,0,15.225),currentpen+1); | ||
+ | draw(graph(f,2,4)^^graph(f,6,10)^^graph(f,12,14),red+currentpen+2); | ||
+ | real dpt=2; | ||
+ | real ts=0.75; | ||
+ | transform st=scale(ts); | ||
+ | label(rotate(90)*st*"Elevation (meters)",(-dpt,8)); | ||
+ | label(st*"Time (seconds)",(8,-dpt)); | ||
+ | </asy> | ||
+ | The requested time intervals are: | ||
* from the <math>2</math>nd to the <math>4</math>th seconds | * from the <math>2</math>nd to the <math>4</math>th seconds | ||
Line 18: | Line 73: | ||
==Solution 2== | ==Solution 2== | ||
+ | Notice that the entire section between the <math>2</math> second mark and the <math>14</math> second mark is between the <math>4</math> and <math>7</math> feet elevation level except the <math>2</math> seconds where she skis just under the <math>4</math> feet mark and when she skis just above the <math>7</math> feet mark, making the answer <math>14-2-2-2=\boxed{\textbf{(B)}\ 8}.</math> | ||
− | + | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | |
+ | https://youtu.be/wQd1lHtkVPU | ||
− | + | ~Education the Study of everything | |
− | == Video Solution | + | ==Video Solution by Math-X (Let's first Understand the question)== |
+ | https://youtu.be/Ku_c1YHnLt0?si=_0SCHsHavl1dJJpP&t=1364 | ||
+ | |||
+ | ~Math-X | ||
+ | |||
+ | == Video Solution by SpreadTheMathLove== | ||
https://www.youtube.com/watch?v=lfyg5ZMV0gg | https://www.youtube.com/watch?v=lfyg5ZMV0gg | ||
+ | |||
+ | https://www.youtube.com/watch?v=TAa6jarbATE | ||
==Video Solution by Magic Square== | ==Video Solution by Magic Square== | ||
https://youtu.be/-N46BeEKaCQ?t=4903 | https://youtu.be/-N46BeEKaCQ?t=4903 | ||
+ | |||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/DBqko2xATxs&t=786 | ||
+ | |||
+ | ==Video Solution by harungurcan== | ||
+ | https://www.youtube.com/watch?v=oIGy79w1H8o&t=15s | ||
+ | |||
+ | ~harungurcan | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2023|num-b=8|num-a=10}} | {{AMC8 box|year=2023|num-b=8|num-a=10}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 04:20, 24 January 2024
Contents
Problem
Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between and
meters?
Solution 1
We mark the time intervals in which Malaika's elevation is between and
meters in red, as shown below:
The requested time intervals are:
- from the
nd to the
th seconds
- from the
th to the
th seconds
- from the
th to the
th seconds
In total, Malaika spends seconds at such elevation.
~apex304, MRENTHUSIASM
Solution 2
Notice that the entire section between the second mark and the
second mark is between the
and
feet elevation level except the
seconds where she skis just under the
feet mark and when she skis just above the
feet mark, making the answer
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education the Study of everything
Video Solution by Math-X (Let's first Understand the question)
https://youtu.be/Ku_c1YHnLt0?si=_0SCHsHavl1dJJpP&t=1364
~Math-X
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=lfyg5ZMV0gg
https://www.youtube.com/watch?v=TAa6jarbATE
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=4903
Video Solution by Interstigation
https://youtu.be/DBqko2xATxs&t=786
Video Solution by harungurcan
https://www.youtube.com/watch?v=oIGy79w1H8o&t=15s
~harungurcan
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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.