Difference between revisions of "2020 AMC 10B Problems/Problem 13"
(→Solution 4 (Answer Choices)) |
MRENTHUSIASM (talk | contribs) (→Solution 3 (Pattern)) |
||
Line 57: | Line 57: | ||
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ||
− | |||
− | |||
− | |||
− | |||
== Solution 4 (Answer Choices) == | == Solution 4 (Answer Choices) == |
Latest revision as of 00:42, 31 October 2024
Contents
Problem
Andy the Ant lives on a coordinate plane and is currently at facing east (that is, in the positive -direction). Andy moves unit and then turns left. From there, Andy moves units (north) and then turns left. He then moves units (west) and again turns left. Andy continues his progress, increasing his distance each time by unit and always turning left. What is the location of the point at which Andy makes the th left turn?
Solution 1 (Associative Property)
Andy makes a total of moves: horizontal ( left and right) and vertical ( up and down).
The -coordinate of Andy's final position is The -coordinate of Andy's final position is Together, we have
~MRENTHUSIASM
Solution 2 (Pattern)
You can find that every four moves both coordinates decrease by Therefore, both coordinates need to decrease by two times. You subtract, giving you the answer of
~happykeeper
Solution 3 (Pattern)
Let's first mark the first few points Andy will arrive at:
Starting Point ( move):
move:
move:
move:
move:
move:
move:
move:
In the move Andy lands on , in the move, Andy lands on .
There is a pattern, for every moves (starting from the move), Andy will arrive on a coordinate in the form of . From this we can deduce:
move:
move:
move:
move:
move:
We have , for which and . So, on the move Andy is at .
Because the problem asks for the move, , on the move, Andy will be on .
Solution 4 (Answer Choices)
Drawing the spiral out, you obtain that every 4th turn occurs on the bottom left corner, whose coordinates difference stay the same(40):
(-20, 20) (-22, 18) (-24, 16) ...
So the X and Y coordinates will always have a difference of 40.
Now looking at the answer choices, only answer choice B's X and Y coordinates have a difference of 40, there (B) is the answer. :)
~Alan
Video Solution (HOW TO CRITICALLY THINK!!!)
~Education, the Study of Everything
Video Solution
https://youtu.be/t6yjfKXpwDs?t=413
Similar Problem
2015 AMC 10B Problem 24 https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_24
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.