A barge travels at 7.5 mph in water. The barge travels 65 miles up the river and 65 miles down the river in a total of 17.75

Question

What's the average speed of current mph in the River in nearest tenth?

Answer 1

D=r*t

 

let x = rate of the river

let t= time up river

 

up the river

65 =(7.5 - x)t >> 65=7.5t - xt >> 65=t(7.5 - x) >> t =65/(7.5 - x)

plug this result into down river equation.

 

down river

65 = (7.5 + x)(17.75 -t)  >> 65 =(7.5 + x)(17.75 -65/(7.5 - x))

 

65  = 133.125 -487.5/(7.5 -x) +17.75x -65x/(7.5 -x)

-68.125 = (-487.5 - 65x)/(7.5 -x) +17.75x

multiply both sides by (7.5 - x)

-68.125(7.5 - x) = -487.5 - 65x +17.75(7.5 - x)

-510.9375 +68.125x =-487.5 - 65x + 133.125 - 17.75x

150.875x =156.5625

x=1.0 mph

t = 10 hrs up river

 

Whew!

 

 

Answer 2

If the barge is traveling at 7.5 mph, it's really not traveling at that speed. It is either getting a boost from traveling with the current or is going slower by traveling against the current.....and we're looking for the influence of the current. Lets call that c. 

Now set up a table like any other motion problem,

 

                      rate                    *         time                    =        distance

with current     7.5 + c                         65/(7.5+c)                          65

against current   7.5-c                          65/(7.5-c)                           65

 

Note: if r*t=d, then d/r = t

 

So we've got 65/(7.5+c) + 65/(7.5-c) = 17.75 for an equation.

multiply both sides by the common denominator of (7.5+c)(7-5-c) to get

 

65(7.5-c) + 65(7.5+c) = 17.75(7.5-c)(7.5+c)     I hope you like the quadratic formula!!!!

487.5 - 65c + 487.5 + 65c = 998.4375 - 17.75c²

975                                  = 998.4375 - 17.75c²

17.75c²                            =23.4375 

c²                                    = 1.320422535    take the square root to get

c                                      =  1.149096399

 

So to the nearest tenth is 1.1 mph for the current.

 

Check it

With the current you actually travel at 8.6 so your time is 65/8.6 = 7.558139535

against current you actually travel at 6.4 so your time is 65/6.4 = 10.15625

Add them up for a total time of 17.714 hrs. It's a little low because we rounded down. If we used 1.15 for the currents speed instead, it would have come out a lot closer.