Difference between revisions of "2023 AMC 8 Problems/Problem 13"
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finish lines. The <math>3</math>rd water station is located <math>2</math> miles after the <math>1</math>st repair station. How long is the race | finish lines. The <math>3</math>rd water station is located <math>2</math> miles after the <math>1</math>st repair station. How long is the race | ||
in miles? | in miles? | ||
| − | + | ||
| + | <asy> | ||
| + | //Diagram By TheMathGuyd, The road is widened for bicycle safety | ||
| + | size(10cm); | ||
| + | real tl = 0.2; | ||
| + | real dpth = 0.8; | ||
| + | fill((0,0)--(8,0)--(8,1)--(0,1)--cycle,mediumgrey); | ||
| + | draw((0,.5)--(8,.5),dashed+white); | ||
| + | draw((0,0)--(8,0)--(8,1)--(0,1)--cycle,black); | ||
| + | draw((1,0)--(1,-tl)); | ||
| + | draw((2,0)--(2,-tl)); | ||
| + | draw((7,0)--(7,-tl)); | ||
| + | label(rotate(45)*"Water 1",(0.6,-dpth)); | ||
| + | label(rotate(45)*"Water 2",(1.6,-dpth)); | ||
| + | label(rotate(45)*"Water 7",(6.6,-dpth)); | ||
| + | label("\ldots\ldots",(4.2,-dpth)); | ||
| + | label(rotate(90)*"Start",(0,0.5),W); | ||
| + | label(rotate(-90)*"Finish",(8,0.5),E); | ||
| + | //Someone is going to see this and find out I don't know how to put borders on text and laugh. | ||
| + | real lh = 0.45; | ||
| + | real lr = 0.1; | ||
| + | path bx = (-lh,-lr)--(0,-lr)--(0,1+lr)--(-lh,1+lr)--cycle; | ||
| + | draw(bx); | ||
| + | draw(reflect((4,0),(4,1))*bx); | ||
| + | </asy> | ||
| + | |||
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96</math> | <math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96</math> | ||
Revision as of 11:55, 26 January 2023
Contents
Problem
Along the route of a bicycle race, 
water stations are evenly spaced between the start and finish lines,
as shown in the figure below. There are also 
repair stations evenly spaced between the start and
finish lines. The 
rd water station is located miles after the 
st repair station. How long is the race
in miles?
![[asy] //Diagram By TheMathGuyd, The road is widened for bicycle safety size(10cm); real tl = 0.2; real dpth = 0.8; fill((0,0)--(8,0)--(8,1)--(0,1)--cycle,mediumgrey); draw((0,.5)--(8,.5),dashed+white); draw((0,0)--(8,0)--(8,1)--(0,1)--cycle,black); draw((1,0)--(1,-tl)); draw((2,0)--(2,-tl)); draw((7,0)--(7,-tl)); label(rotate(45)*"Water 1",(0.6,-dpth)); label(rotate(45)*"Water 2",(1.6,-dpth)); label(rotate(45)*"Water 7",(6.6,-dpth)); label("\ldots\ldots",(4.2,-dpth)); label(rotate(90)*"Start",(0,0.5),W); label(rotate(-90)*"Finish",(8,0.5),E); //Someone is going to see this and find out I don't know how to put borders on text and laugh. real lh = 0.45; real lr = 0.1; path bx = (-lh,-lr)--(0,-lr)--(0,1+lr)--(-lh,1+lr)--cycle; draw(bx); draw(reflect((4,0),(4,1))*bx); [/asy]](http://latex.artofproblemsolving.com/e/d/d/edd55467fc82abbe5109d8bdb4c20106bd995b20.png)
Solution
Suppose that the race is 
miles long. The water stations are located at ![\[\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}\]](http://latex.artofproblemsolving.com/5/d/7/5d75992c71e35e3d504ae9cbb3b9aeeb572d0fbb.png)
miles from the start, and the repair stations are located at ![\[\frac{d}{3}, \frac{2d}{3}\]](http://latex.artofproblemsolving.com/b/d/c/bdc9e9aa2443a5c15309d91a1a860ab82d30094a.png)
miles from the start.
We are given that 
from which
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM
Video Solution (Animated)
~Star League (https://starleague.us)
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=4439
See Also
| 2023 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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. 
