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Difference between revisions of "2021 Fall AMC 10A Problems/Problem 11"

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==Problem==
 
==Problem==
  
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Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster tha the ship. She counts <math>210</math> equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts <math>42</math> steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
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Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts <math>210</math> equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts <math>42</math> steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
  
 
<math>\textbf{(A) }70\qquad\textbf{(B) }84\qquad\textbf{(C) }98\qquad\textbf{(D) }105\qquad\textbf{(E) }126</math>
 
<math>\textbf{(A) }70\qquad\textbf{(B) }84\qquad\textbf{(C) }98\qquad\textbf{(D) }105\qquad\textbf{(E) }126</math>
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==Solution 1==
 
==Solution 1==
 
Let the speed at which Emily walks be <math>42</math> steps per hour. Let the speed at which the ship is moving be <math>s</math>. Walking in the direction of the ship, it takes her <math>210</math> steps, or <math>\frac {210}{42} = 5</math> hours, to travel. We can create the equation: <cmath>d = 5(42-s)</cmath> Where d is the length of the ship. Walking in the opposite direction of the ship, it takes her 42 steps, or 42/42 = 1 hour. We can create a similar equation: <cmath>d = 1(42+s)</cmath> Now we have 2 variables and 2 equations, and we can solve for d.  
 
Let the speed at which Emily walks be <math>42</math> steps per hour. Let the speed at which the ship is moving be <math>s</math>. Walking in the direction of the ship, it takes her <math>210</math> steps, or <math>\frac {210}{42} = 5</math> hours, to travel. We can create the equation: <cmath>d = 5(42-s)</cmath> Where d is the length of the ship. Walking in the opposite direction of the ship, it takes her 42 steps, or 42/42 = 1 hour. We can create a similar equation: <cmath>d = 1(42+s)</cmath> Now we have 2 variables and 2 equations, and we can solve for d.  

Revision as of 18:02, 24 November 2021

Problem

Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?

$\textbf{(A) }70\qquad\textbf{(B) }84\qquad\textbf{(C) }98\qquad\textbf{(D) }105\qquad\textbf{(E) }126$

Solution 1

Let the speed at which Emily walks be $42$ steps per hour. Let the speed at which the ship is moving be $s$. Walking in the direction of the ship, it takes her $210$ steps, or $\frac {210}{42} = 5$ hours, to travel. We can create the equation: \[d = 5(42-s)\] Where d is the length of the ship. Walking in the opposite direction of the ship, it takes her 42 steps, or 42/42 = 1 hour. We can create a similar equation: \[d = 1(42+s)\] Now we have 2 variables and 2 equations, and we can solve for d. \[210-5s = 42 + s\] \[s = 28\] \[d = 42 + s = \boxed{\textbf{(A) } 70}\] ~LucaszDuzMatz

~Minor LaTeX edits by Arcticturn

Solution 2

Let $x$ be the length of the ship. Then, in the time that Emily walks $210$ steps, the ship moves $210-x$ steps. Also, in the time that Emily walks $42$ steps, the ship moves $x-42$ steps. Since the ship and Emily both travel at some constant rate, $\frac{210}{210-x} = \frac{42}{x-42}$. Dividing both sides by $42$ and cross multiplying, we get $5(x-42) = 210-x$, so $6x = 420$, and $x = \boxed{70}$. ~ihatemath123

See Also

2021 Fall AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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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