Difference between revisions of "2000 AMC 10 Problems/Problem 18"
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draw( (0,0)--(-1,0), dashed ); draw( (0,0)--(0,-1), dashed ); | draw( (0,0)--(-1,0), dashed ); draw( (0,0)--(0,-1), dashed ); | ||
draw( (5,0)--(6,0), dashed ); draw( (5,0)--(5,-1), dashed ); | draw( (5,0)--(6,0), dashed ); draw( (5,0)--(5,-1), dashed ); | ||
| − | draw( (5,5)--(6,5), dashed ); | + | draw( (5,5)--(6,5), dashed ); // This line is correct |
| − | + | draw( (5,5)--(5,6), dashed ); // Add semicolon here | |
| − | |||
draw( (0,5)--(-1,5), dashed ); draw( (0,5)--(0,6), dashed ); | draw( (0,5)--(-1,5), dashed ); draw( (0,5)--(0,6), dashed ); | ||
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Therefore the total area she can see is <math>16 + 4\cdot 5 + \pi\cdot 1^2 = 36+\pi \simeq 39.14</math>, which rounded to the nearest integer is <math>39</math>. <math>\boxed{C}</math> | Therefore the total area she can see is <math>16 + 4\cdot 5 + \pi\cdot 1^2 = 36+\pi \simeq 39.14</math>, which rounded to the nearest integer is <math>39</math>. <math>\boxed{C}</math> | ||
| + | |||
| + | ==Video Solution by Daily Dose of Math== | ||
| + | |||
| + | https://youtu.be/509okdgEaFM?si=ns2w-utssCI_6GJx | ||
| + | |||
| + | ~Thesmartgreekmathdude | ||
==See Also== | ==See Also== | ||
Latest revision as of 23:10, 20 July 2025
Problem
Charlyn walks completely around the boundary of a square whose sides are each 
km long. From any point on her path she can see exactly 
km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
Video Solution
https://youtu.be/j7Hi5I8INII - Happytwin
Solution
The area she sees looks at follows:
![[asy] unitsize(0.8cm); path p1 = (0,0)--(5,0)--(5,5)--(0,5)--cycle; path p2 = (1,1)--(4,1)--(4,4)--(1,4)--cycle; path p3 = arc((0,0),1,180,270) -- arc((5,0),1,270,360) -- arc((5,5),1,0,90) -- arc((0,5),1,90,180) -- cycle; fill(p3,lightgray); unfill(p2); draw(p1,linewidth(bp)); draw(p2); draw(p3); draw( (0,0)--(-1,0), dashed ); draw( (0,0)--(0,-1), dashed ); draw( (5,0)--(6,0), dashed ); draw( (5,0)--(5,-1), dashed ); draw( (5,5)--(6,5), dashed ); // This line is correct draw( (5,5)--(5,6), dashed ); // Add semicolon here draw( (0,5)--(-1,5), dashed ); draw( (0,5)--(0,6), dashed ); draw( (0,-1)--(5,-1), Arrows ); label( "$5$", (2.5,-1), S ); draw( (1,1)--(4,1), Arrows ); label( "$3$", (2.5,1), N ); draw( (4,3.5)--(5,3.5), Arrows ); label( "$1$", (4.5,3.5), N ); draw( (5,3.5)--(6,3.5), Arrows ); label( "$1$", (5.5,3.5), N ); [/asy]](http://latex.artofproblemsolving.com/2/1/4/214ed07f00ea903b469f5a7ca917777b856e8be4.png)
The part inside the walk has area 
. The part outside the walk consists of four rectangles, and four arcs. Each of the rectangles has area 
. The four arcs together form a circle with radius .
Therefore the total area she can see is 
, which rounded to the nearest integer is 
. 
Video Solution by Daily Dose of Math
https://youtu.be/509okdgEaFM?si=ns2w-utssCI_6GJx
~Thesmartgreekmathdude
See Also
| 2000 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 AMC 10 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
