Difference between revisions of "2021 Fall AMC 10A Problems/Problem 11"
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<cmath>\begin{align*} | <cmath>\begin{align*} | ||
210-5s &= 42 + s \\ | 210-5s &= 42 + s \\ | ||
− | s &= 28 \\ | + | s &= 28. \\ |
\end{align*}</cmath> | \end{align*}</cmath> | ||
Therefore, we have <math>d = 42 + s = \boxed{\textbf{(A) }70}</math>. | Therefore, we have <math>d = 42 + s = \boxed{\textbf{(A) }70}</math>. |
Revision as of 13:16, 26 November 2021
Contents
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 equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts
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?
Solution 1 (One Variable)
Let be the length of the ship.
Then, in the time that Emily walks
steps, the ship moves
steps.
Also, in the time that Emily walks
steps, the ship moves
steps.
Since the ship and Emily both travel at some constant rate,
. Dividing both sides by
and cross multiplying, we get
, so
, and
.
~ihatemath123
Solution 2 (Two Variables)
Let the speed at which Emily walks be steps per hour. Let the speed at which the ship is moving be
. Walking in the direction of the ship, it takes her
steps, or
hours, to travel. We can create an equation:
where
is the length of the ship. Walking in the opposite direction of the ship, it takes her
steps, or
hour. We can create a similar equation:
Now we have two variables and two equations. We can equate the expressions for
and solve for
:
Therefore, we have
.
~LucaszDuzMatz (Solution)
~Arcticturn (Minor Edits)
Solution 3 (Three Variables)
Denote by the length of the ship,
and
the rates of Emily and the ship (distance per walking step), respectively.
Hence, and
.
By solving these equations, we get .
Plugging this result into the first equation, we get
.
This is exactly the length of the ship, measured in terms of Emily's equal steps.
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.