Word problem models involving rational equations

Question

A boat which moves at 18 miles per hour in still water travels 14 miles downstream in the same amount of time it takes to travel 10 miles upstream. Find the speed of the current.

 

 

 

Please help and show work so I can learn thank you 

Answer 1

boat moves at 18 miles per hour in still water. 18 miles per hour equals velocity, and it's the velocity of the boat, so I'll call it v_b

v_b = 18m/h            

v of boat downstream = v_d   

v of boat upstream = v_u   

distance upstream = d_u = 10 miles = 10m

distance downstream = d_d = 14 miles = 14m

time = t          you must know that distance = velocity * time!

 

we want to know the velocity of the current, which = v_c

v_d = v_b + v_c because the current pushes the boat along

v_u = v_b - v_c because the current pushes against the boat

d_u = v_u*t    =      (v_b - v_c)t     = 10m                             d_d = v_d*t =  (v_b + v_c)t  = 14m

 

14m = 10m + 4m

 

so

 

d_d = d_u + 4m

 

so

 

d_d = (v_b - v_c)t + 4m

 

(v_b + v_c)t = (v_b - v_c)t + 4m

 

so, we know v_b = 18m/h

 

(18m/h + v_c)t = (18m/h - v_c)t + 4m

 

18mt/h + v_ct = 18mt/h - v_ct + 4m

 

subtract 18mt/h from both sides and you get:

 

v_ct = -v_ct + 4m

 

add -v_ct to both sides

 

2v_ct = 4m

 

v_c = 2m/t               so now we need to figure out what t equals

 

(vb + vc)t = 14m

(18m/h + 2m/t)t = 14m

 

18mt/h + 2m = 14m

 

18mt/h = 12m

 

t = (12m/18m)h = 2h/3

 

v_c = 2m/t = 2m/(2h/3) = 2m*3/2h = 3m/h = 3 miles per hour

 

whew! that's a lot of calculations!