Difference between revisions of "2021 AMC 10A Problems/Problem 4"

Revision as of 07:01, 23 February 2021

Problem

A cart rolls down a hill, travelling $5$ inches the first second and accelerating so that during each successive $1$ -second time interval, it travels $7$ inches more than during the previous $1$ -second interval. The cart takes $30$ seconds to reach the bottom of the hill. How far, in inches, does it travel?

$\textbf{(A)} ~215 \qquad\textbf{(B)} ~360\qquad\textbf{(C)} ~2992\qquad\textbf{(D)} ~3195\qquad\textbf{(E)} ~3242$

Solution 1 (Arithmetic Series)

Since $\text{Distance}=\text{Speed}\times\text{Time},$ we seek the sum $5(1)+12(1)+19(1)+26(1)+\cdots=5+12+19+26+\cdots,$ in which there are $30$ addends. The last addend is $5+7(30-1)=208.$ Therefore, the requested sum is $5+12+19+26+\cdots+208=\frac{(5+208)(30)}{2}=\boxed{\textbf{(D)} ~3195}.$ 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 $\frac{\text{First}+\text{Last}}{2}\cdot\text{Count}.$ ~MRENTHUSIASM

Solution 2 (Answer Choices and Modular Arithmetic)

From the $30$ -term sum $5+12+19+26+\cdots$ in the previous solution, taking modulo $10$ gives $5+12+19+26+\cdots \equiv 3(0+1+2+3+4+5+6+7+8+9) = 3(45)\equiv5 \pmod{10}.$ The only answer choices that are $5\mod{10}$ are $\textbf{(A)}$ and $\textbf{(D)}.$ By a quick estimation, $\textbf{(A)}$ is too small, leaving us with $\boxed{\textbf{(D)} ~3195}.$

~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)

https://youtu.be/7NSfDCJFRUg

~ pi_is_3.14

Video Solution (Simple and Quick)

https://youtu.be/qLDkSnxLvxM

~ Education, the Study of Everything

Video Solution 4

https://youtu.be/aO-GklwkBfI

~savannahsolver

Video Solution by TheBeautyofMath

https://youtu.be/50CThrk3RcM?t=262

~IceMatrix

@@ Line 8: / Line 8: @@
 ==Solution 2 (Answer Choices and Modular Arithmetic)==
-From the <math>30</math>-term sum <cmath>5+12+19+26+\cdots</cmath> in the previous solution, taking modulo <math>10</math> gives <cmath>5+12+19+26+\cdots \equiv 3(0+1+2+3+4+5+6+7+8+9) = 3(45)\equiv5 \pmod{10}.</cmath> The only answer choices that are <math>5\mod{10}</math> are <math>\textbf{(A)}</math> and <math>\textbf{(D)}.</math> By a quick estimate, <math>\textbf{(A)}</math> is too small, leaving us with <math>\boxed{\textbf{(D)} ~3195}.</math> ~MRENTHUSIASM
+From the <math>30</math>-term sum <cmath>5+12+19+26+\cdots</cmath> in the previous solution, taking modulo <math>10</math> gives <cmath>5+12+19+26+\cdots \equiv 3(0+1+2+3+4+5+6+7+8+9) = 3(45)\equiv5 \pmod{10}.</cmath> The only answer choices that are <math>5\mod{10}</math> are <math>\textbf{(A)}</math> and <math>\textbf{(D)}.</math> By a quick estimation, <math>\textbf{(A)}</math> is too small, leaving us with <math>\boxed{\textbf{(D)} ~3195}.</math>
+~MRENTHUSIASM
 == Video Solution (Arithmetic Sequence but in a different way)==

2021 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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