SOLUTION: It takes 32 hours for a motorboat moving downriver to get from pier A to pier B. The return journey takes 48 hours. How long does it take an unpowered raft to cover this distance?

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Question 1029944: It takes 32 hours for a motorboat moving downriver to get from pier A to pier B. The return journey takes 48 hours. How long does it take an unpowered raft to cover this distance?
Found 2 solutions by josmiceli, ikleyn:
Answer by josmiceli(19441)   (Show Source): You can put this solution on YOUR website!
It takes longer one way than the other
way because there is a current
Let = the distance between A and B.
The units do not matter, since no units
of distance are specified
Let = the speed of the current in units of 1/1 units/hr
Let = the speed of the boat with no current in units of 1/1 units/hr
Note that 1 unit of distance is 1/100 of the distance AB
Let = time in hrs for raft floating with
the current to travel between A and B
--------------------------------------
The motorboat going with the current:
(1)
The motorboat going against the current:
(2)
The raft drifting WITH the current:
(3)
-----------------------
There are 3 equations and 3 unknowns,
so it's should be solvable
(1)
(2)
-----------------------
Divide both sides of (1) by
(1)
Multiply both sides of (1) by
(1)
----------------------------
Add (1) and (2)
(1)
(2)
---------------------

units/hr
and
(1)
(1)
(1)
(1)
(1)
(1) units/hr
-------------------------
The raft drifting WITH the current:
(3)
(3)
(3)
(3)
The unpowered raft would take 192 hrs
to floor with the current between A and B
-----------------
check the answer:
(1)
(1)
(1)
(1)
( error due to rounding off )
(2)
(2)
(2)
(2)
( error due to rounding off )
(3)
(3)
(3)
( error due to rounding off )
OK

Answer by ikleyn(52794)   (Show Source): You can put this solution on YOUR website!
.
It takes 32 hours for a motorboat moving downriver to get from pier A to pier B. The return journey takes 48 hours.
How long does it take an unpowered raft to cover this distance?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Raft (see this article of Wikipedia)
   A raft is any flat structure for support or transportation over water. It is the most basic of boat design, 
   characterized by the absence of a hull. Although there are cross-over boat types that blur this definition, 
   rafts are usually kept afloat by using any combination of buoyant materials such as wood, sealed barrels, 
   or inflated air chambers (such as pontoons), and are typically not propelled by an engine.

It is assumed in this problem that the unpowered raft moves along the river with the current' speed.
Therefore, all we need to do is to determine the speed of the current.

Let D be the distance along the river from pier A to pier B.

Let "u" be the motorboat speed in still water and "v" be the current speed.

When moving upstream, the motorboat has the speed  u-v  (relative to the river's banks),  and it is equal to  .
When moving downstream, the motorboat has the speed  u+v  (relative to the river's banks),  and it is equal to  .
So, you have these two equations

u + v = ,   (1)
u - v = .   (2)

Distract equation (2) from equation (1)  (both sides).  You will get

2v =  =  =  = .

Hence,  v =  = .

Now, the time for the unpowered raft to cover the distance D is 

 =  = 192 hours.

Answer.  The time for the unpowered raft to get from pier A to pier B is 192 hours.


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