linear eqns.?

Question

A boat takes 6 hrs to travel 16 km upstream and 24 km downstream, and it takes 13hrs to travel 36km upstream and 48km downstream. find the speed of the boat in still water and the speed of the stream.

Answer 1

Let "x" the speed of the boat in still water
and "y" the speed of the stream.
Then the total speed upstream is x - y.
and the total speed downstream is x + y.

Let t be the time going upstream the first time.
the boat travels downstream for 6-t hours. { coz the total time going both ways is 6 hours,}


So the speed going up is distance/time = 16/t and going down is 24/(6-t). This gives us:
x - y = 16/t
x + y = 24/(6-t)

Likewise, if "T" is the time going up the stream the second time, we have:
x - y = 36/T
x + y = 48/(13-T)

solving these we get the required answers

Answer 2

Maybe I'm missing something, but those appear to be two separate problems. There is enough information contained in "A boat takes 6 hrs to travel 16 km upstream and 24 km downstream" to solve for both pieces of information

Distance = Time X Velocity
16 = 6(B - S)
24 = 6(B + S)
where B is the velocity of the boat
and S is the velocity of the stream

Adding the two yields
40 = 12B
B = 3.3333

24 = 6*3.3333 + 6S
4 = 6S
S = 0.6666

The second set of equations gives
36 = 13(B - S)
48 = 13(B + S)

84 = 26B
B = 3.2308

48 = 13*3.2308 + 13S
6 = 13S
S = 0.4615

If you add all of the equations, then you get B = 3.2632. If you then substitute that value back into equation set 1 you get S = 0.7368, but if you substitute into equation set 2 you get S = 0.4291. So the best that you can do is to minimize the error in a least squares or similar sense.

Answer 3

Let "B" the speed of the boat in still water and "S" the speed of the stream. Then the total speed upstream is B - S. Likewise, the total speed downstream is B + S.

Let t be the time going upstream the first time. Since the total time going both ways is 6 hours, the boat travels downstream for 6-t hours. So the speed going up is distance/time = 16/t and going down is 24/(6-t). This gives us:
B - S = 16/t
B + S = 24/(6-t)

Likewise, if "u" is the time going up the stream the second time, we have:
B - S = 36/u
B + S = 48/(13-u)

You've got 4 equations with 4 unknowns, so solve for B and S.

Answer 4

Let x= speed in still water and y = speed of current
You then get the equations
16/(x-y)+24/(x+y)=6
36/(x-y)+48/(x+y)=13
Multiply each through by (x-y)(x+y) and solve each for (x-y)(x+y).
Set the results equal: (40x-8y)/6=(84x-12y)/13
Solve to find x=2y.
Substitute in either equation above to find x=8 and y=4