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On a recent Car Talk Show there recently was a puzzle from the "Hat in the River" series. The hosts of the show emphasized that the puzzle should be solved by thinking and not by using algebra which may confuse things.
So instead of thinking I tried to use algebra to try to solve the puzzle. I tried to get it down to two equations and two unknowns and it will be easy. So I thought, but I need help.
Here is the puzzle.
THE PROBLEM:
A vacationer decides to rent a rowboat and go for a ride on a river. He rows upstream for one mile and his hat falls in the river and the river's current carries the hat away from him towards the dock where he started. He continues rowing for 10 more minutes(0.1667 hours) in the same direction.
He instantly turns around after 10 minutes and rows to retrieve the hat which has flowed away from him. He rows with the same effort downstream as he rowed upstream. The hat and the boat arrive at the dock at the same time.
What was the speed of the current?
ATTEMPT AT A SOLUTION:
Vb=speed of boat(no current)
Vc= speed of current
Vb+Vc=speed of boat downstream
Vb-Vc=speed of boat upstrem
Some equations: T(h)=time for hat to reach the dock=1 mile /Vc or simply 1/Vc
T(b)=time for boat to reach the dock. T(b) is the sum of three parts of the trip
First part: 10 minutes(0.1667 hours)he continued rowing upstream after he lost his hat.
Second part: 0.1667x(Vb-Vc)/(Vb+Vc) is the time it takes the rower to row from the turnaround to where is hat fell in the water.This is downstream.
Third part: 1 mile/(Vb+Vc) = time it takes to complete the last mile to the dock.
We know: T(h)=T(b) so
1/Vc = 0.1667+.1667x(Vb-Vc)/(Vb+Vc) +1/(Vb+Vc)
Now I am stuck and I need help.I have one equation and two unknowns(Vb,Vc).
I need one more equation with these variables. Anybody have any ideas for the second equation.
So instead of thinking I tried to use algebra to try to solve the puzzle. I tried to get it down to two equations and two unknowns and it will be easy. So I thought, but I need help.
Here is the puzzle.
THE PROBLEM:
A vacationer decides to rent a rowboat and go for a ride on a river. He rows upstream for one mile and his hat falls in the river and the river's current carries the hat away from him towards the dock where he started. He continues rowing for 10 more minutes(0.1667 hours) in the same direction.
He instantly turns around after 10 minutes and rows to retrieve the hat which has flowed away from him. He rows with the same effort downstream as he rowed upstream. The hat and the boat arrive at the dock at the same time.
What was the speed of the current?
ATTEMPT AT A SOLUTION:
Vb=speed of boat(no current)
Vc= speed of current
Vb+Vc=speed of boat downstream
Vb-Vc=speed of boat upstrem
Some equations: T(h)=time for hat to reach the dock=1 mile /Vc or simply 1/Vc
T(b)=time for boat to reach the dock. T(b) is the sum of three parts of the trip
First part: 10 minutes(0.1667 hours)he continued rowing upstream after he lost his hat.
Second part: 0.1667x(Vb-Vc)/(Vb+Vc) is the time it takes the rower to row from the turnaround to where is hat fell in the water.This is downstream.
Third part: 1 mile/(Vb+Vc) = time it takes to complete the last mile to the dock.
We know: T(h)=T(b) so
1/Vc = 0.1667+.1667x(Vb-Vc)/(Vb+Vc) +1/(Vb+Vc)
Now I am stuck and I need help.I have one equation and two unknowns(Vb,Vc).
I need one more equation with these variables. Anybody have any ideas for the second equation.