a grain barge travels on a river from point a to point b loading and unloading grain. the barge travels at a rate of 5mph relative to the water. the rive flows downstream at a rate of 1mph. if the trip upstream takes 2 hours longer than the trip downstream, how far is it from point a to point b?
a grain barge travels on a river from point a to point b loading and unloading grain.
Mathematically the given information is as follows:
Time up stream: Tu = Td + 2 (1) Tu & Td are up &down stream times
Distance A to B: D = Tu * Su (2) Su is speed upstream
Distance B to A: D = Td * Sd (3) D is the same up or down stream!
Su is speed upstream: Su = 5 -1 = 4 mph (4) barge is slowed upstream
Sd is speed down stream: Sd = 5 +1 = 6 mph (5 ) barge moves faster downstream
From (2) & (3) find:
Tu*Su = Td*Sd
Tu*Su/Sd = Td
Tu*4/6 = Td
Td = 2/3*Tu (6) Makes sense…Td<Tu
Substitute (6) in (1) and isolate Tu:
Tu = Td +2
Tu= 2/3*Tu +2
Tu – 2/3* Tu = 2
1/3*Tu = 2
Tu = 3*2
Tu = 6
Now, substitute Tu=6 into (1) and isolate Td:
Tu =Td + 2
6= Td +2
6-2 = Td
Td = 4
Find D by substituting Sd and Td results into (3)
D = 4*6
D= 24 miles
Check this D result by substituting Tu & Su into (3):
D = Tu*Su
D= 6 * 4
D=24 miles OK!