Difference between revisions of "2023 AMC 8 Problems/Problem 9"
(→Solution 1) |
Themathguyd (talk | contribs) (→Problem: Replace picture with asy, making sure to use the actual cubic to get the same graph) |
||
| Line 1: | Line 1: | ||
==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)); | ||
| + | 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> | ||
Revision as of 16:25, 25 January 2023
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?
![[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)); 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]](http://latex.artofproblemsolving.com/2/2/7/2277e87f963256a7bfc6d3fd227250fe0bc30cea.png)
Solution
The time intervals in which Malaika's elevation is between and meters are:
- from the
nd to the 
th seconds
nd to the - from the
th to the 
th seconds
th to the 
th seconds- from the
th to the 
th seconds
th to the 
th secondsIn total, Malaika spends 
seconds at such elevation.
~apex304, MRENTHUSIASM
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=lfyg5ZMV0gg
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=4903
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. 
