Difference between revisions of "2022 AMC 8 Problems/Problem 10"
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==Problem== | ==Problem== | ||
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| + | One sunny day, Ling decided to take a hike in the mountains. She left her house at <math>8 \, \textsc{pm}</math>, drove at a constant speed of <math>45</math> miles per hour, and arrived at the hiking trail at <math>10 \, \textsc{am}</math>. After hiking for <math>3</math> hours, Ling drove home at a constant speed of <math>60</math> miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip? | ||
| + | |||
| + | [asy] | ||
| + | unitsize(12); | ||
| + | usepackage("mathptmx"); | ||
| + | defaultpen(fontsize(8)+linewidth(.7)); | ||
| + | int mod12(int i) {if (i<13) {return i;} else {return i-12;}} | ||
| + | void drawgraph(pair sh,string lab) { | ||
| + | for (int i=0;i<11;++i) { | ||
| + | for (int j=0;j<6;++j) { | ||
| + | draw(shift(sh+(i,j))*unitsquare,mediumgray); | ||
| + | } | ||
| + | } | ||
| + | draw(shift(sh)*((-1,0)--(11,0)),EndArrow(angle=20,size=8)); | ||
| + | draw(shift(sh)*((0,-1)--(0,6)),EndArrow(angle=20,size=8)); | ||
| + | for (int i=1;i<10;++i) { | ||
| + | draw(shift(sh)*((i,-.2)--(i,.2))); | ||
| + | } | ||
| + | label("8\tiny{\textsc{am}}",sh+(1,-.2),S); | ||
| + | |||
| + | for (int i=2;i<9;++i) { | ||
| + | label(string(mod12(i+7)),sh+(i,-.2),S); | ||
| + | } | ||
| + | label("4\tiny{\textsc{pm}}",sh+(9,-.2),S); | ||
| + | for (int i=1;i<6;++i) { | ||
| + | label(string(30*i),sh+(0,i),2*W); | ||
| + | } | ||
| + | draw(rotate(90)*"Distance (miles)",sh+(-2.1,3),fontsize(10)); | ||
| + | label("\$\textbf{("+lab+")}\$",sh+(-2.1,6.8),fontsize(10)); | ||
| + | } | ||
| + | drawgraph((0,0),"A"); | ||
| + | drawgraph((15,0),"B"); | ||
| + | drawgraph((0,-10),"C"); | ||
| + | drawgraph((15,-10),"D"); | ||
| + | drawgraph((0,-20),"E"); | ||
| + | dotfactor=6; | ||
| + | draw((1,0)--(3,3)--(6,3)--(8,0),linewidth(.9)); | ||
| + | dot((1,0)^^(3,3)^^(6,3)^^(8,0)); | ||
| + | pair sh = (15,0); | ||
| + | draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(8,0)),linewidth(.9)); | ||
| + | dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(8,0)); | ||
| + | pair sh = (0,-10); | ||
| + | draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(7.5,0)),linewidth(.9)); | ||
| + | dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(7.5,0)); | ||
| + | pair sh = (15,-10); | ||
| + | draw(shift(sh)*((1,0)--(3,4)--(6,4)--(9.3,0)),linewidth(.9)); | ||
| + | dot(sh+(1,0)^^sh+(3,4)^^sh+(6,4)^^sh+(9.3,0)); | ||
| + | pair sh = (0,-20); | ||
| + | draw(shift(sh)*((1,0)--(3,3)--(6,3)--(7.5,0)),linewidth(.9)); | ||
| + | dot(sh+(1,0)^^sh+(3,3)^^sh+(6,3)^^sh+(7.5,0)); | ||
| + | [/asy] | ||
==Solution== | ==Solution== | ||
Revision as of 12:00, 28 January 2022
Problem
One sunny day, Ling decided to take a hike in the mountains. She left her house at 
, drove at a constant speed of 
miles per hour, and arrived at the hiking trail at 
. After hiking for 
hours, Ling drove home at a constant speed of 
miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip?
[asy] unitsize(12); usepackage("mathptmx"); defaultpen(fontsize(8)+linewidth(.7)); int mod12(int i) {if (i<13) {return i;} else {return i-12;}} void drawgraph(pair sh,string lab) { for (int i=0;i<11;++i) { for (int j=0;j<6;++j) { draw(shift(sh+(i,j))*unitsquare,mediumgray); } } draw(shift(sh)*((-1,0)--(11,0)),EndArrow(angle=20,size=8)); draw(shift(sh)*((0,-1)--(0,6)),EndArrow(angle=20,size=8)); for (int i=1;i<10;++i) { draw(shift(sh)*((i,-.2)--(i,.2))); } label("8\tiny{\textsc{am}}",sh+(1,-.2),S);
for (int i=2;i<9;++i) { label(string(mod12(i+7)),sh+(i,-.2),S); } label("4\tiny{\textsc{pm}}",sh+(9,-.2),S); for (int i=1;i<6;++i) { label(string(30*i),sh+(0,i),2*W); } draw(rotate(90)*"Distance (miles)",sh+(-2.1,3),fontsize(10)); label("$\textbf{("+lab+")}$",sh+(-2.1,6.8),fontsize(10)); } drawgraph((0,0),"A"); drawgraph((15,0),"B"); drawgraph((0,-10),"C"); drawgraph((15,-10),"D"); drawgraph((0,-20),"E"); dotfactor=6; draw((1,0)--(3,3)--(6,3)--(8,0),linewidth(.9)); dot((1,0)^^(3,3)^^(6,3)^^(8,0)); pair sh = (15,0); draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(8,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(8,0)); pair sh = (0,-10); draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(7.5,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(7.5,0)); pair sh = (15,-10); draw(shift(sh)*((1,0)--(3,4)--(6,4)--(9.3,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,4)^^sh+(6,4)^^sh+(9.3,0)); pair sh = (0,-20); draw(shift(sh)*((1,0)--(3,3)--(6,3)--(7.5,0)),linewidth(.9)); dot(sh+(1,0)^^sh+(3,3)^^sh+(6,3)^^sh+(7.5,0)); [/asy]
Solution
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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. 
