Determine the speed of the current

Question

A chartered riverboat cruise travels 48 miles upstream, then returns to port. The boat speed without a current is 20 mph. If th round trip takes 5 hours, what is the speed of the current? 

Answer 1

distance = rate x time

 

48 miles = (20 - X)* (t)  <----  X is the speed of the current and t is the time going upstream (against the current)

 

              = (20 + x)*(5-t)

 

  20t - xt = 100 - 20t + 5x - xt <--- distributive on the left; FOIL on the right

 

 20t = 100 - 20t + 5x  <--- adds xt to both sides

 

  40t = 100 + 5x <--- adds 20t to both sides 

 

 8t = 20 + x <--- divides everything by 5

 

x = 8t - 20

 

 

48 = (20 - x)t

 

     = (20 - (8t-20))t  <--- substitutes x = 8t - 20

 

      = ( 20 - 8t + 20)t

 

      = (40 - 8t)t

 

48 = 40t - 8t^2

 

 

6 = 5t - t^2 <--- divides everything by 8

 

t^2 - 5t + 6 = 0

 

( t    -    3)(t    -     2  )  = 0

 

 t -3 = 0 ---> t = 3

t-2 = 0 ---> t=2

 

t=2 must be the time traveling downstream with the current

So it took the boat 3 hours to travel upstream against the current.

 

48 = (20 - X)*3

 

16 = 20 -x

 

16-20 = -x

 

-4 = -x

 

x=4

 

The speed of the current is 4 mph

 

CHECK:

  Upstream (20 - 4)*3 = 16*3 = 48

  Downstream (20 + 4)*2 = 24 * 2 = 48