River or lake?

Logic Level 2

This problem can be solved logically or algebraically:

You are at a camp and your friend is injured. There are two ways to get to a doctor: cross the river or cross the lake. The river and the lake are both the same distance. The river has a current. The lake has no current. Which route should you take in order to get to the doctor and back in the shortest amount of time? (Assume that you row faster than the current of the river.)

Lake River Equally fast

Blcraft Gaming by BLCRAFT Gaming , 15, USA

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2 solutions

Blcraft Gaming
Feb 22, 2018

The logical explanation is that going on the river, you will be hurt by the current for a longer time than you will be helped by the current. Therefore, the lake would be the faster method.

You could also imagine a scenario in which your speed going against the current is 1 1 mph or something like that. Obviously in that case, the lake would be faster.

Somebody else can post the algebraic solution below :)

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i thought you were going with thecurrent

NSCS 747 - 10 months, 1 week ago

Let d be the distance, s the speed of your rowing and c the speed of the current. Thus, the total round trip time , T is the sum of the times spent travelling with, and then against the current. Algebraically, T= d/(s+c)+d/(s-c)=(d(s-c)+d(s+c))/(s+c)(s-c)=2sd/s^2-c^2=2sd/(s^2(1-c^2/s^2))=2d/s[1/(1-(c/s)^2]

When there is no current, c=0. then T=2d/s. When c>0, the denominator in the bracket gets smaller and thus T> 2d/s. A current will always increase the total travel time.

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