Difference between revisions of "2013 AMC 12B Problems/Problem 11"
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Thus they reach 10 feet away when A is moving east and B is moving west, between moves 7 and 8. Thus the answer is <math>\textbf{(A)}</math> | Thus they reach 10 feet away when A is moving east and B is moving west, between moves 7 and 8. Thus the answer is <math>\textbf{(A)}</math> | ||
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+ | ==Solution 2== | ||
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+ | Denote the origin of the two bees as A. Let the final positions of the bees be B and C respectively. Since they are moving an equal distance away from the origin with each of their respective steps, two right angled triangles can be constructed from their final positions. Since these triangles are congruent, they each have a hypotenuse of five units so they are both (3,4,5) triangles. 'Resolving' these triangles into the moves of each bee shows that they must have taken 7 steps to reach their final position at which point Bee B must move east next and Bee C is moving west. Hence <math>\textbf{(A)}</math> | ||
== See also == | == See also == | ||
{{AMC12 box|year=2013|ab=B|num-b=10|num-a=12}} | {{AMC12 box|year=2013|ab=B|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 15:35, 27 November 2019
Contents
Problem
Two bees start at the same spot and fly at the same rate in the following directions. Bee travels foot north, then foot east, then foot upwards, and then continues to repeat this pattern. Bee travels foot south, then foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly feet away from each other?
east, west
north, south
north, west
up, south
up, west
Solution
Let A and B begin at (0,0,0). In 6 steps, A will have done his route twice, ending up at (2,2,2), and B will have done his route three times, ending at (-3,-3,0). Their distance is We now move forward one step at a time until they are ten feet away: 7 steps: A moves north to (2,3,2), B moves south to (-3,-4,0), distance of 8 steps: A moves east to (3,3,2), B moves west to (-4,-4,0), distance of
Thus they reach 10 feet away when A is moving east and B is moving west, between moves 7 and 8. Thus the answer is
Solution 2
Denote the origin of the two bees as A. Let the final positions of the bees be B and C respectively. Since they are moving an equal distance away from the origin with each of their respective steps, two right angled triangles can be constructed from their final positions. Since these triangles are congruent, they each have a hypotenuse of five units so they are both (3,4,5) triangles. 'Resolving' these triangles into the moves of each bee shows that they must have taken 7 steps to reach their final position at which point Bee B must move east next and Bee C is moving west. Hence
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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 12 Problems and Solutions |
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