Difference between revisions of "2025 AMC 8 Problems/Problem 5"
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| − | + | ==Problem== | |
| + | Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. | ||
| + | Betty starts at the factory (labled <math>F</math>) and drives to location <math>A</math>, then <math>B</math>, then <math>C</math>, before returning to <math>F</math>. What is the shortest distance, in blocks, she can drive to complete the route? | ||
| + | |||
| + | <asy> | ||
| + | |||
| + | unitsize(20); | ||
| + | |||
| + | add(grid(8,6)); | ||
| + | |||
| + | path w = circle((0,0),0.4); | ||
| + | |||
| + | fill(w, white); | ||
| + | draw(w); | ||
| + | label("$B$",(0,0)); | ||
| + | |||
| + | fill(shift((2,4)) * w, white); | ||
| + | draw(shift((2,4)) * w); | ||
| + | label("$C$",(2,4)); | ||
| + | |||
| + | fill(shift((7,3)) * w, white); | ||
| + | draw(shift((7,3)) * w); | ||
| + | label("$A$",(7,3)); | ||
| + | |||
| + | fill(shift((6,5)) * w, white); | ||
| + | draw(shift((6,5)) * w); | ||
| + | label("$F$",(6,5)); | ||
| + | |||
| + | draw((6,-0.2)--(7,-0.2), EndArrow(3)); | ||
| + | draw((7,-0.2)--(6,-0.2), EndArrow(3)); | ||
| + | draw(shift(6.5, -0.48) * scale(0.03) * texpath("1 block")); | ||
| + | |||
| + | draw((8.2,1)--(8.2,2), EndArrow(3)); | ||
| + | draw((8.2,2)--(8.2,1), EndArrow(3)); | ||
| + | draw(shift(8.88, 1.5) * scale(0.03) * texpath("1 block")); | ||
| + | |||
| + | </asy> | ||
| + | |||
| + | <math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 22 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 26\qquad \textbf{(E)}\ 28</math> | ||
| + | |||
| + | ==Solution 1== | ||
| + | |||
| + | Each shortest possible path from <math>A</math> to <math>B</math> follows the edges of the rectangle. The following path outlines a path of <math>\boxed{24}</math> units: | ||
| + | |||
| + | <asy> | ||
| + | |||
| + | unitsize(20); | ||
| + | |||
| + | add(grid(8,6)); | ||
| + | draw((6,5)--(7,5)--(7,0)--(0,0)--(0,4)--(2,4)--(2,5)--cycle,green); | ||
| + | |||
| + | path w = circle((0,0),0.4); | ||
| + | |||
| + | fill(w, white); | ||
| + | draw(w); | ||
| + | label("$B$",(0,0)); | ||
| + | |||
| + | fill(shift((2,4)) * w, white); | ||
| + | draw(shift((2,4)) * w); | ||
| + | label("$C$",(2,4)); | ||
| + | |||
| + | fill(shift((7,3)) * w, white); | ||
| + | draw(shift((7,3)) * w); | ||
| + | label("$A$",(7,3)); | ||
| + | |||
| + | fill(shift((6,5)) * w, white); | ||
| + | draw(shift((6,5)) * w); | ||
| + | label("$F$",(6,5)); | ||
| + | |||
| + | </asy> | ||
| + | |||
| + | ~ [[zhenghua]] | ||
| + | |||
| + | ==Video Solution by Daily Dose of Math== | ||
| + | |||
| + | https://youtu.be/rjd0gigUsd0 | ||
| + | |||
| + | ~Thesmartgreekmathdude | ||
| + | |||
| + | ==Video Solution by Thinking Feet== | ||
| + | https://youtu.be/PKMpTS6b988 | ||
| + | |||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2025|num-b=4|num-a=6}} | {{AMC8 box|year=2025|num-b=4|num-a=6}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 20:24, 30 January 2025
Contents
Problem
Betty drives a truck to deliver packages in a neighborhood whose street map is shown below.
Betty starts at the factory (labled 
) and drives to location 
, then 
, then 
, before returning to . What is the shortest distance, in blocks, she can drive to complete the route?
![[asy] unitsize(20); add(grid(8,6)); path w = circle((0,0),0.4); fill(w, white); draw(w); label("$B$",(0,0)); fill(shift((2,4)) * w, white); draw(shift((2,4)) * w); label("$C$",(2,4)); fill(shift((7,3)) * w, white); draw(shift((7,3)) * w); label("$A$",(7,3)); fill(shift((6,5)) * w, white); draw(shift((6,5)) * w); label("$F$",(6,5)); draw((6,-0.2)--(7,-0.2), EndArrow(3)); draw((7,-0.2)--(6,-0.2), EndArrow(3)); draw(shift(6.5, -0.48) * scale(0.03) * texpath("1 block")); draw((8.2,1)--(8.2,2), EndArrow(3)); draw((8.2,2)--(8.2,1), EndArrow(3)); draw(shift(8.88, 1.5) * scale(0.03) * texpath("1 block")); [/asy]](http://latex.artofproblemsolving.com/0/2/f/02f4d70dd8e934b785c58b9e171db52b9bca463e.png)
Solution 1
Each shortest possible path from to follows the edges of the rectangle. The following path outlines a path of 
units:
![[asy] unitsize(20); add(grid(8,6)); draw((6,5)--(7,5)--(7,0)--(0,0)--(0,4)--(2,4)--(2,5)--cycle,green); path w = circle((0,0),0.4); fill(w, white); draw(w); label("$B$",(0,0)); fill(shift((2,4)) * w, white); draw(shift((2,4)) * w); label("$C$",(2,4)); fill(shift((7,3)) * w, white); draw(shift((7,3)) * w); label("$A$",(7,3)); fill(shift((6,5)) * w, white); draw(shift((6,5)) * w); label("$F$",(6,5)); [/asy]](http://latex.artofproblemsolving.com/8/a/d/8ad8c099dd8bd66bb4ff7b18df7d0eee48fccb36.png)
~ zhenghua
Video Solution by Daily Dose of Math
~Thesmartgreekmathdude
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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. 
