Difference between revisions of "2011 AMC 10B Problems/Problem 12"
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− | == Problem | + | == Problem== |
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second? | Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second? | ||
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===Solution 2=== | ===Solution 2=== | ||
− | It is basically the same as Solution 1 except you can completely disregard the straight edges of the track since it will take Keiko the same time to walk that length. Instead, | + | It is basically the same as Solution 1 except you can completely disregard the straight edges of the track since it will take Keiko the same time to walk that length for both inner and outer circles. Instead, focus on the circular part. If the diameter of the smaller circle were <math>r,</math> then the length of the smaller circle would be <math>r \pi</math> and the length of the larger circle would be <math>(r+12) \pi.</math> Since it still takes <math>36</math> seconds longer, |
<cmath> \begin{align*} | <cmath> \begin{align*} | ||
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s&=\boxed{\textbf{(A)} \frac{\pi}{3}} | s&=\boxed{\textbf{(A)} \frac{\pi}{3}} | ||
\end{align*} </cmath> | \end{align*} </cmath> | ||
+ | |||
+ | == See Also== | ||
+ | |||
+ | {{AMC10 box|year=2011|ab=B|num-b=11|num-a=13}} | ||
+ | {{MAA Notice}} |
Latest revision as of 15:15, 24 August 2021
Problem
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of meters, and it takes her
seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
Solution
Solution 1
Let be Keiko's speed in meters per second,
be the length of the straight parts of the track,
be the radius of the smaller circles, and
be the radius of the larger circles. The length of the inner edge will be
and the length of the outer edge will be
Since it takes
seconds longer for Keiko to walk on the outer edge,
Solution 2
It is basically the same as Solution 1 except you can completely disregard the straight edges of the track since it will take Keiko the same time to walk that length for both inner and outer circles. Instead, focus on the circular part. If the diameter of the smaller circle were then the length of the smaller circle would be
and the length of the larger circle would be
Since it still takes
seconds longer,
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 |
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