2021 Fall AMC 10A Problems/Problem 11
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 have the same ratio of absolute speeds in either direction, . Dividing both sides by and cross multiplying, we get , so , and .
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 .
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 be the length of the ship, be Emily's step length, and be the ship's step length. We wish to find
When Emily walks from the back of the ship to the front, she walks a distance of and the front of the ship moves a distance of We have for this scenario, which rearranges to When Emily walks in the opposite direction, she walks a distance of and the back of the ship moves a distance of We have for this scenario, which rearranges to We multiply by and then add to get from which
~Steven Chen (www.professorchenedu.com)
Solution 4 (Using the Boat's "Step")
Every time Emily takes a step, the boat also "takes a step". Call the length of the boats step . Call the length of the boat .
When Emily is walking in the same direction as the boat, every time she takes a step the boat moves an additional distance of . This means that she travels a total distance of to reach the other end of the boat.
When Emily is walking in the opposite direction of the boat, every time she takes a step the distance till the end of the boat reduces by (since the boat is coming towards her and moves a distance of ). This means that she travels a total distance of to reach the other end of the boat.
Taking Emily's step as a unit of distance, we now have two equations Solving for you get .
Solution 5 (Relative Speeds)
Call the speed of the boat and the speed of Emily .
Consider the scenario when Emily is walking along with the boat. Relative to an observer on the boat, her speed is .
Consider the scenario when Emily is walking in the opposite direction. Relative to an observer on the boat, her speed is
Since Emily takes steps to walk along with the boat and steps to walk opposite the boat, that means it takes her times longer to walk the length of a stationary boat at compared to .
This means that , so .
As Emily takes steps to walk the length of the boat at a speed of , she must take of the time to walk the length of the boat at a speed of , so our answer is .
Video Solution by TheBeautyofMath
Video Solution by WhyMath
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