Problema Solution
Gabriel paddled a canoe 5 miles upstream at a constant speed of 15 mph. He then turned around and paddled downstream until he reached his original starting place. If the entire trip took him 9 hours, how fast was the speed of the river current?
Answer provided by our tutors
let
v = the canoes speed in still water, v>0
c = speed of the current
d = 5 miles the distance he traveled upstream
downstream speed: v + c
t1 = the time traveling downstream
upstream speed: v - c = 15
t2 = the time traveling upstream
the entire trip took him 9 hours
t1 + t2 = 9 =>t1 = 9 -t2
Gabriel paddled a canoe 5 miles upstream at a constant speed of 15 mph
v - c = 15
since speed = distance/time
downstream:
v + c = 5/t1
upstream:
v - c = 5/t2
since v - c = 15 we have 5/t2 = 15 => t2 = 5/15 = 1/3 hours
t2 = 1/3
t1 = 9 - t2 = 9 - 1/3 = 26/3 hours
notice that the downstream time is bigger then the upstream time t1 > t2 and this should not happen if he traveled the same distance upstream and downstream. This will give us a result with a negative speed of the river current.
lets plug t1 and t2 in the equations for v and c:
v + c = 5/(26/3)
v + c = 15/26
v - c = 15
by solving the system of equations we find and consider only the positive roots since v>0
v = 7.79 mph
c = -7.2 mph
click here to see the step by step solution of the system of equations
indeed we got negative value for the river current meaning that something must be wrong in the text of the problem. Perhaps Gabriel paddled a canoe 5 miles downstream instead of upstream at a constant speed of 15 mph?