downstream/upstream word problem

Question

A boat travels downstream and then travels upstream back to the original starting point. If the total time going upstream and downstream is 2 hours, find the exact speed of boat in still water and the approximate the speed of the boat using 1 decimal place accuracy. Assume the current is 3mph.

Answer 1

distance = rate x time

 

  D = R * T = RT

 

downstream (with the current)

    distance D = (r+3)*T <--- current speeds up the boat by 3 mph faster; T is the time to get there.

                                                

 

  upstream (AGAINST the current)

  distance D =(r-3)*(2-t) <--- current slows down the boat by 3 mph slower;

                                                the total round trip took 2 hours; so the time is (2-t)

 

 The distances are the same:

  (r + 3)* t = (r-3)*(2-t)

 

  rt + 3t = 2r - 6 - rt + 3t  <--- distributive on the left; FOIL on the right; work over this step carefully!!!

                                                      2 things going on.

 

 rt = 2r - 6 - rt <--- subtracts 3t form both sides

 

 2rt = 2r - 6  <--- adds rt to both sides

 

 2rt = 2(r-3)  <--- factors 2 out of right side

 

rt = r - 3  <--- divides both sides by 2

 

Solving for r:

 

rt - r = -3  <--- moves the r to the left side and factors it out

 

r(t-1) = -3

 

r = -3/(t-1)  <---- divides by t-1

 

r = 3/(1-t)  <--- multiplies the right side by 1 = -1/-1 which changes the signs and causes the bottom to switch

 

We now have the speed of the boat in still water as a function of time.

 

in order for the rate of speed to be positive, 1-t >0; So 1 > t ---> t < 1
so 0 < t < 1 and cannot be equal to 1 or else division by zero.
and t cannot be equal to zero because getting there in zero time is unreal.
So t = 0.1, 0.2, 0.3, 0.4, .... 0.9 because the directions say 1 digit accuracy.

 

The following table summarizes the possible answers:

 

time t downstream                0.1          0.2           0.3           0.4             0.5          0.6          0.7          0.8          0.9
speed of boat in still water   3.33         3.75        4.29             5                6            7.5          10          15            30
speed upstream                 0.33          0.75        1.29             2                3            4.5           7           12            27
speed downstream              6.33         6.75        7.29             8                9           10.5         13          18            33
time upstream                    1.9            1.8          1.7            1.6             1.5          1.4          1.3          1.2           1.1
distance upstream            0.63            1.35       2.19             3.2            4.5          6.3          9.1          14.4        29.7
distance downstream        0.63            1.35       2.19             3.2            4.5          6.3          9.1          14.4        29.7