Difference between revisions of "2021 AMC 10A Problems/Problem 4"
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==Problem== | ==Problem== | ||
A cart rolls down a hill, travelling <math>5</math> inches the first second and accelerating so that during each successive <math>1</math>-second time interval, it travels <math>7</math> inches more than during the previous <math>1</math>-second interval. The cart takes <math>30</math> seconds to reach the bottom of the hill. How far, in inches, does it travel? | A cart rolls down a hill, travelling <math>5</math> inches the first second and accelerating so that during each successive <math>1</math>-second time interval, it travels <math>7</math> inches more than during the previous <math>1</math>-second interval. The cart takes <math>30</math> seconds to reach the bottom of the hill. How far, in inches, does it travel? | ||
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~ pi_is_3.14 | ~ pi_is_3.14 | ||
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+ | ==Video Solution (Simple and Quick)== | ||
+ | https://youtu.be/qLDkSnxLvxM | ||
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+ | Education, the Study of Everything | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2021|ab=A|num-b=3|num-a=5}} | {{AMC10 box|year=2021|ab=A|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 15:34, 15 February 2021
Contents
Problem
A cart rolls down a hill, travelling inches the first second and accelerating so that during each successive -second time interval, it travels inches more than during the previous -second interval. The cart takes seconds to reach the bottom of the hill. How far, in inches, does it travel?
Solution 1
Since we seek the sum in which there are 30 addends. The last addend is Therefore, the requested sum is Recall that to find the sum of an arithmetic series, we take the average of the first and last terms, then multiply by the number of terms, namely ~MRENTHUSIASM
Solution 2 (Answer Choices and Modular Arithmetic)
From the -term sum in the previous solution, taking modulo gives The only answer choices that are are and By a quick estimate, is too small, leaving us with ~MRENTHUSIASM
Video Solution (Arithmetic Sequence but in a different way)
https://www.youtube.com/watch?v=sJa7uB-UoLc&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=4
~ North America Math Contest Go Go Go
Video Solution (Using Arithmetic Sequence)
~ pi_is_3.14
Video Solution (Simple and Quick)
Education, the Study of Everything
See Also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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.