Difference between revisions of "2021 AMC 10A Problems/Problem 4"
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| + | ==Problem 4== | ||
| + | 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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| + | <math>\textbf{(A)} ~215 \qquad\textbf{(B)} ~360\qquad\textbf{(C)} ~2992\qquad\textbf{(D)} ~3195\qquad\textbf{(E)} ~3242</math> | ||
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| + | ==Solution== | ||
| + | Since <cmath>\text{Distance}=\text{Speed}\times\text{Time},</cmath> we seek the sum <cmath>5(1)+12(1)+19(1)+26(1)+\cdots=5+12+19+26+\cdots,</cmath> in which there are 30 addends. The last addend is <math>5+7(30-1)=208.</math> Therefore, the requested sum is <cmath>5+12+19+26+\cdots+208=\frac{(5+208)(30)}{2}=\boxed{\text{(D) } 3195}.</cmath> 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 <cmath>\frac{\text{First}+\text{Last}}{2}\cdot\text{Count}.</cmath> | ||
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== Video Solution (Using Arithmetic Sequence) == | == Video Solution (Using Arithmetic Sequence) == | ||
https://youtu.be/7NSfDCJFRUg | https://youtu.be/7NSfDCJFRUg | ||
~ pi_is_3.14 | ~ pi_is_3.14 | ||
Revision as of 21:33, 11 February 2021
Problem 4
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
Since ![\[\text{Distance}=\text{Speed}\times\text{Time},\]](http://latex.artofproblemsolving.com/6/0/6/606216cd23e3b2f30aad3c379f14bbe7df7522e0.png)
we seek the sum ![\[5(1)+12(1)+19(1)+26(1)+\cdots=5+12+19+26+\cdots,\]](http://latex.artofproblemsolving.com/8/c/7/8c79e9f6d74d79045a0538ba9ba95d45a5f6d918.png)
in which there are 30 addends. The last addend is 
Therefore, the requested sum is ![\[5+12+19+26+\cdots+208=\frac{(5+208)(30)}{2}=\boxed{\text{(D) } 3195}.\]](http://latex.artofproblemsolving.com/a/a/d/aadb9bbead55745590da3bf3ff43a13a2b80085f.png)
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 ![\[\frac{\text{First}+\text{Last}}{2}\cdot\text{Count}.\]](http://latex.artofproblemsolving.com/f/e/6/fe6482f623264044d4444569428eb2cc7d8c0d9f.png)
Video Solution (Using Arithmetic Sequence)
~ pi_is_3.14
