Difference between revisions of "2022 AIME I Problems/Problem 5"
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Since Melanie and Sherry swim for the same distance and the same amount of time, they swim at the same net speed. | Since Melanie and Sherry swim for the same distance and the same amount of time, they swim at the same net speed. | ||
− | + | Let <math>x</math> and <math>y</math> be some positive numbers. We have the following table: | |
+ | <cmath>\begin{array}{c||c|c|c} | ||
+ | & \textbf{Net Velocity Vector (m/min)} & \textbf{Natural Velocity Vector (m/min)} & \textbf{Natural Speed (m/min)} \\ | ||
+ | \hline \hline | ||
+ | &&& \\ [-2.25ex] | ||
+ | \textbf{Melanie} & \langle -x,y\rangle & \langle -x-14,y\rangle & 80 \\ \hline | ||
+ | &&& \\ [-2.25ex] | ||
+ | \textbf{Sherry} & \langle x,y\rangle & \langle x-14,y\rangle & 60 | ||
+ | \end{array}</cmath> | ||
+ | Recall that <math>|\text{velocity}|=\text{speed}.</math> It follows that | ||
+ | <cmath>\begin{align*} | ||
+ | (-x-14)^2 + y^2 &= 80^2, \\ | ||
+ | (x-14)^2 + y^2 &= 60^2. | ||
+ | \end{align*}</cmath> | ||
+ | ~MRENTHUSIASM | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=I|num-b=4|num-a=6}} | {{AIME box|year=2022|n=I|num-b=4|num-a=6}} |
Revision as of 19:14, 17 February 2022
Contents
Problem
A straight river that is meters wide flows from west to east at a rate of
meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of
meters downstream from Sherry. Relative to the water, Melanie swims at
meters per minute, and Sherry swims at
meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find
.
Solution 1
Define as the number of minutes they swam for.
Let their meeting point be . In an alternative reality, there would be no current. Then, had they swum facing the same direction that they had in the real universe, they would've met at a point west of
. Precisely, since the water moves at
meters per minute, this alternative reality meeting point would have been
meters to the left of
.
So, our alternative reality is just a geometry problem now:
Note that while this diagram was drawn knowing the correct dimensions, we do not actually know that the triangle with sides
,
and
is a right triangle yet, so we cannot use that information.
By Pythagorean, we have
Subtracting the first equation from the second gives us , so
. Substituting this into our first equation, we have that
So .
~ ihatemath123
Solution 2
We have the following diagram:
WILL BE AVAILABLE WITHIN THE NEXT DAY.
Since Melanie and Sherry swim for the same distance and the same amount of time, they swim at the same net speed.
Let and
be some positive numbers. We have the following table:
Recall that
It follows that
~MRENTHUSIASM
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |