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

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==Problem==
 
==Problem==
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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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== Solution ==  
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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 1 (Arithmetic Series)==
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Since <cmath>\mathrm{Distance}=\mathrm{Speed}\cdot\mathrm{Time},</cmath> we seek the sum <cmath>5\cdot1+12\cdot1+19\cdot1+26\cdot1+\cdots=5+12+19+26+\cdots,</cmath> in which there are <math>30</math> terms. The last term is <math>5+7\cdot(30-1)=208.</math> Therefore, the requested sum is <cmath>5+12+19+26+\cdots+208=\frac{5+208}{2}\cdot30=\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: <cmath>\mathrm{Sum}=\frac{\mathrm{First}+\mathrm{Last}}{2}\cdot\mathrm{Count}.</cmath> ~MRENTHUSIASM
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==Solution 2 (Answer Choices and Modular Arithmetic)==
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From the <math>30</math>-term sum <cmath>5+12+19+26+\cdots</cmath> in Solution 1, taking modulo <math>10</math> gives <cmath>5+12+19+26+\cdots \equiv 3\cdot(0+1+2+3+4+5+6+7+8+9) = 3\cdot45\equiv5 \pmod{10}.</cmath> The only answer choices congruent to <math>5</math> modulo <math>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>
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~MRENTHUSIASM
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==Solution 3==
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The distance (in inches) traveled within each 1-second interval is:
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<math>5,5+1(7),5+2(7), \dots , 5+29(7).</math>
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This is an arithmetic sequence so the total distance travelled, found by summing them up is:
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<math>\text{number of terms} \cdot \text{average of terms} = \text{number of terms} \cdot \dfrac{\text{first term}+\text{last term}}{2}.</math>
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Or,
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<math>30 \cdot \dfrac{5+5+29(7)}{2} = 15 \cdot 213 = \boxed{\textbf{(D)} ~3195}.</math>
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~BakedPotato66
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==Video Solution (Simple and Quick)==
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https://youtu.be/qLDkSnxLvxM
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~ Education, the Study of Everything
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 +
== Video Solution (Arithmetic Sequence but in a Different Way)==
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https://www.youtube.com/watch?v=sJa7uB-UoLc&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=4
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~ North America Math Contest Go Go Go
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== Video Solution (Using Arithmetic Sequence) ==
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https://youtu.be/7NSfDCJFRUg
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~ pi_is_3.14
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==Video Solution==
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https://youtu.be/aO-GklwkBfI
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~savannahsolver
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==Video Solution by TheBeautyofMath==
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https://youtu.be/50CThrk3RcM?t=262
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~IceMatrix
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==Video Solution by The Learning Royal==
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https://youtu.be/slVBYmcDMOI
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==See Also==
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{{AMC10 box|year=2021|ab=A|num-b=3|num-a=5}}
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{{MAA Notice}}

Revision as of 20:30, 3 July 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 \[\mathrm{Distance}=\mathrm{Speed}\cdot\mathrm{Time},\] we seek the sum \[5\cdot1+12\cdot1+19\cdot1+26\cdot1+\cdots=5+12+19+26+\cdots,\] in which there are $30$ terms. The last term is $5+7\cdot(30-1)=208.$ Therefore, the requested sum is \[5+12+19+26+\cdots+208=\frac{5+208}{2}\cdot30=\boxed{\textbf{(D)} ~3195}.\] 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: \[\mathrm{Sum}=\frac{\mathrm{First}+\mathrm{Last}}{2}\cdot\mathrm{Count}.\] ~MRENTHUSIASM

Solution 2 (Answer Choices and Modular Arithmetic)

From the $30$-term sum \[5+12+19+26+\cdots\] in Solution 1, taking modulo $10$ gives \[5+12+19+26+\cdots \equiv 3\cdot(0+1+2+3+4+5+6+7+8+9) = 3\cdot45\equiv5 \pmod{10}.\] The only answer choices congruent to $5$ modulo $10$ are $\textbf{(A)}$ and $\textbf{(D)}.$ By a quick estimation, $\textbf{(A)}$ is too small, leaving us with $\boxed{\textbf{(D)} ~3195}.$

~MRENTHUSIASM

Solution 3

The distance (in inches) traveled within each 1-second interval is:

$5,5+1(7),5+2(7), \dots , 5+29(7).$

This is an arithmetic sequence so the total distance travelled, found by summing them up is: $\text{number of terms} \cdot \text{average of terms} = \text{number of terms} \cdot \dfrac{\text{first term}+\text{last term}}{2}.$

Or,

$30 \cdot \dfrac{5+5+29(7)}{2} = 15 \cdot 213 = \boxed{\textbf{(D)} ~3195}.$

~BakedPotato66

Video Solution (Simple and Quick)

https://youtu.be/qLDkSnxLvxM

~ Education, the Study of Everything

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

https://youtu.be/aO-GklwkBfI

~savannahsolver

Video Solution by TheBeautyofMath

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

~IceMatrix

Video Solution by The Learning Royal

https://youtu.be/slVBYmcDMOI

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

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