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

Revision as of 08:21, 12 February 2021

Problem 4

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

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 $5+12+19+26+\cdots \equiv 3(1+2+3+\cdots+10) = 3(55)\equiv5 \pmod{10}.$ The only answer choices that are $5\mod{10}$ are $\textbf{(A)}$ and $\textbf{(D)}.$ By a quick estimate, $\textbf{(A)}$ is too small, leaving us with $\boxed{\textbf{(D)} ~3195}.$ ~MRENTHUSIASM

Video Solution (Using Arithmetic Sequence)

https://youtu.be/7NSfDCJFRUg

~ pi_is_3.14

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.

@@ Line 7: / Line 7: @@
 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{\textbf{(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> ~MRENTHUSIASM
-==Solution 2 (Answer Choices)==
+==Solution 2 (Answer Choices and Modular Arithmetic)==
-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. Taking modulo <math>10,</math> we get <cmath>5+12+19+26+\cdots \equiv 3(1+2+3+\cdots+10) = 3(55)\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(1+2+3+\cdots+10) = 3(55)\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
 == Video Solution (Using Arithmetic Sequence) ==

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