Difference between revisions of "2008 AMC 8 Problems/Problem 18"
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| − | ==Problem== | + | == Problem == |
Two circles that share the same center have radii <math>10</math> meters and <math>20</math> meters. An aardvark runs along the path shown, starting at <math>A</math> and ending at <math>K</math>. How many meters does the aardvark run? | Two circles that share the same center have radii <math>10</math> meters and <math>20</math> meters. An aardvark runs along the path shown, starting at <math>A</math> and ending at <math>K</math>. How many meters does the aardvark run? | ||
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<asy> | <asy> | ||
size((150)); | size((150)); | ||
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label("$K$", (20,0), E); | label("$K$", (20,0), E); | ||
</asy> | </asy> | ||
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<math> \textbf{(A)}\ 10\pi+20\qquad\textbf{(B)}\ 10\pi+30\qquad\textbf{(C)}\ 10\pi+40\qquad\textbf{(D)}\ 20\pi+20\qquad \\ \textbf{(E)}\ 20\pi+40</math> | <math> \textbf{(A)}\ 10\pi+20\qquad\textbf{(B)}\ 10\pi+30\qquad\textbf{(C)}\ 10\pi+40\qquad\textbf{(D)}\ 20\pi+20\qquad \\ \textbf{(E)}\ 20\pi+40</math> | ||
| − | ==See Also== | + | == Solution == |
| + | We will deal with this part by part: | ||
| + | Part 1: 1/4 circumference of big circle= <math>\frac{2\pi r}{4}=\frac{\pi r}{2}=\frac{20\pi}{2}=10\pi</math> | ||
| + | Part 2: Big radius minus small radius= <math>20-10=10</math> | ||
| + | Part 3: 1/4 circumference of small circle= <math>\frac{\pi r}{2}=\frac{10\pi}{2}=5\pi</math> | ||
| + | Part 4: Diameter of small circle: <math>2*10=20</math> | ||
| + | Part 5: Same as part 3: <math>5\pi</math> | ||
| + | Part 6: Same as part 2: <math>10</math> | ||
| + | Total: <math>10\pi + 10 + 5\pi + 20 + 5\pi + 10 = \boxed{E = 20\pi + 40}</math> | ||
| + | |||
| + | == Video Solution == | ||
| + | https://www.youtube.com/watch?v=NJs4rFyXFwQ | ||
| + | —DSA_Catachu | ||
| + | |||
| + | == See Also == | ||
{{AMC8 box|year=2008|num-b=17|num-a=19}} | {{AMC8 box|year=2008|num-b=17|num-a=19}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 12:37, 19 October 2020
Contents
Problem
Two circles that share the same center have radii 
meters and 
meters. An aardvark runs along the path shown, starting at 
and ending at 
. How many meters does the aardvark run?
![[asy] size((150)); draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle); draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle); draw((20,0)--(-20,0)); draw((0,20)--(0,-20)); draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow); draw((-18,1)--(-12, 1), EndArrow); draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow); draw((1,-9)--(1,9), EndArrow); draw((0,12)..(8.3, 8.3)..(12,0), EndArrow); draw((12,-1)--(18,-1), EndArrow); label("$A$", (0,20), N); label("$K$", (20,0), E); [/asy]](http://latex.artofproblemsolving.com/2/1/0/210387e06e9467a3ab329cbdf9e43ff63316fb45.png)
Solution
We will deal with this part by part:
Part 1: 1/4 circumference of big circle= 
Part 2: Big radius minus small radius= 
Part 3: 1/4 circumference of small circle= 
Part 4: Diameter of small circle: 
Part 5: Same as part 3: 
Part 6: Same as part 2:
Total: 
Video Solution
https://www.youtube.com/watch?v=NJs4rFyXFwQ —DSA_Catachu
See Also
| 2008 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
