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

Revision as of 00:23, 28 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 (One Variable)

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 $x = \boxed{\textbf{(A) }70}$ .

~ihatemath123

Solution 2 (Two Variables)

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 an 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 two variables and two equations. We can equate the expressions for $d$ and solve for $s$ : $\begin{align*} 210-5s &= 42 + s \\ s &= 28. \\ \end{align*}$ Therefore, we have $d = 42 + s = \boxed{\textbf{(A) }70}$ .

~LucaszDuzMatz (Solution)

~Arcticturn (Minor $\LaTeX$ Edits)

Solution 3 (Three Variables)

Suppose that Emily and the ship take steps simultaneously such that Emily's steps cover a greater length than the ship's steps.

Let $L$ be the length of the ship, $E$ be Emily's step length, and $S$ be the ship's step length. We wish to find $\frac LE.$

When Emily walks from the back of the ship to the front, she walks a distance of $210E$ and the front of the ship moves a distance of $210S.$ We have $210E=L+210S$ for this scenario, which rearranges to $210E-210S=L. \hspace{15mm}(1)$ When Emily walks in the opposite direction, she walks a distance of $42E$ and the back of the ship moves a distance of $42S.$ We have $42E=L-42S$ for this scenario, which rearranges to $42E+42S=L. \hspace{19.125mm}(2)$ We multiply $(2)$ by $5$ and then add $(1)$ to the result: $420E=6L,$ from which $\frac LE = \boxed{\textbf{(A) }70}.$

~Steven Chen (www.professorchenedu.com)

~MRENTHUSIASM

Solution 4 (Relative Speeds)

Call the speed of the boat $v_s$ and the speed of Emily $v_e$ . Consider the scenario when Emily is walking along with the boat. Relative to an observer on the boat, her speed is $v_e-v_s$ . Consider the scenario when Emily is walking in the opposite direction. Relative to an observer on the boat, her speed is $v_e+v_s$ Since Emily takes $210$ steps to walk along with the boat and $42$ steps to walk opposite the boat, that means it takes her $5$ times longer to walk the length of a stationary boat at $v_e-v_s$ compared to $v_e+v_s$ . This means that $5(v_e-v_s)=v_e+v_s \rightarrow v_s = \frac{2v_e}{3}$ As Emily takes $210$ steps to walk the length of the boat at a speed of $v_e- \frac{2v_e}{3}=\frac{v_e}{3}$ , she must take $\frac13$ of the time to walk the length of the boat at a speed of $v_e$ , so our answer is $210/3 \rightarrow \boxed{\textbf{(A) }70}.$

@@ Line 41: / Line 41: @@
 Consider the scenario when Emily is walking along with the boat. Relative to an observer on the boat, her speed is <math>v_e-v_s</math>.
 Consider the scenario when Emily is walking in the opposite direction. Relative to an observer on the boat, her speed is <math>v_e+v_s</math>
-Since Emily takes <math>210</math> steps to walk along with the boat and <math>42</math> steps to walk opposite the boat, that means it takes her 5x longer to walk the length of a stationary boat at <math>v_e-v_s</math> compared to <math>v_e+v_s</math>.
+Since Emily takes <math>210</math> steps to walk along with the boat and <math>42</math> steps to walk opposite the boat, that means it takes her <math>5</math> times longer to walk the length of a stationary boat at <math>v_e-v_s</math> compared to <math>v_e+v_s</math>.
 This means that <math>5(v_e-v_s)=v_e+v_s \rightarrow v_s = \frac{2v_e}{3}</math>
-As Emily takes <math>210</math> steps to walk the length of the boat at a speed of <math>v_e- \frac{2v_e}{3}=\frac{v_e}{3}</math>, she must take 1/3rd of the time to walk the length of the boat at a speed of <math>v_e</math>, so our answer is <math>210/3 \rightarrow \boxed{\textbf{(A) }70}.</math>
+As Emily takes <math>210</math> steps to walk the length of the boat at a speed of <math>v_e- \frac{2v_e}{3}=\frac{v_e}{3}</math>, she must take <math>\frac13</math> of the time to walk the length of the boat at a speed of <math>v_e</math>, so our answer is <math>210/3 \rightarrow \boxed{\textbf{(A) }70}.</math>
 ==See Also==
 {{AMC10 box|year=2021 Fall|ab=A|num-b=10|num-a=12}}
 {{MAA Notice}}

2021 Fall AMC 10A (Problems • Answer Key • Resources)
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