Word problems

Question

motorboat maintained a constant speed of 24 miles per hour relative to the water in going 16 miles upstream and then returning. The total time for the trip was 1.5 hours. Use the information to find the speed of the current

Answer 1

Use Distance = rate * time : d= r*t

Upstream : boat is against the water so speed goes down : speed of the current = - r

downstream : boat is flowing with the water so speed become faster   speed of the current =  +r

 

time is 1.5 together 

 

create table

 

___________________d____________r__________t_

upstream                      16                   24-r                   t                 

Downstream                  16                   24+r              1.5 - t

 

Write out equations using d= r*t for both up and down stream

 

16 = (24-r) *t

16 = (24+r)*(1.5-t)

 

2 system of eq use substitution:

 

16 / (24-r) = t    solve for t in the 1st equation 

16 = (24+r)*(1.5 - (16 / (24-r) ) )  substitute for t in the 2nd equation

 

work with the inside brackets: 1.5 - (16 / (24-r) )  find LCD  (24-r) and  simplify the fraction by Multiplying LCD to 1.5

 

1.5 (24-r)  / (24-r) - 16 / (24-r)  distribute 1.5 t the bracket and simplify

 

(36 -1.5r -16) / (24-r) 

 

16 = (24+r)*( 36- 1.5r -16 / (24-r))    Combine like terms

 

16 = (24+r)*( 36- 1.5r -16 / (24-r))    Combine like terms

16 = (24+r)*( 20- 1.5r  / (24-r))   rewrite the equation
16 = (24+r)*( 20- 1.5r )/(24-r)  Cross multiply the denominator

16 (24-r) = (24+r)*( 20- 1.5r )  distribute the 16 and solve

384 -16r = (24+r)( 20- 1.5r )  FOIL the right hand side

384 -16r = 480 - 36r +20r -1.5r^2  Combine like terms

384 -16r = 480 - 16r  -1.5r^2  move r and number over

-96 = -1.5 r^2  divide

64 = r^2  apply square root

 

r = 8 (only positive answer since it is a speed of the current)

 

 

 

 

 

 

 

   

rewrite the equation
rewrite the equation

Answer 2

v = d/t

--> t = d/v  (this is the velocity equation rearranged to solve for t)

 

Total time = time upstream + time downstream

1.5 hours = d/v (upstream) + d/v (downstream)

1.5 hours = 16 miles/ v_upstream + 16 miles/ v_downstream

 

*v_upstream = traveling velocity - speed of current

*v_downstream = traveling velocity + speed of current

 

The way I read the question, the motorboat is going 24 mph relative to the water, so I would say that the traveling velocity is 24 mph.  We can let the speed of the current be represented by x in our equation.

 

1.5 hours = 16 miles / (24 - x) + 16 miles / (24 + x)

 

From here, we can use what we know about solving equations (get a common denominator, cancel factors, distribute, get x by itself) to find the speed of the current.

 

Common denominator: (24 - x)(24 + x)

 

1.5 (24 - x)(24 + x) = 16 (24 + x) + 16 (24 - x)

1.5 (576 - x²) = 384 + 16x + 384 - 16x

1.5 (576 - x²) = 768

576 - x² = 512

- x² = -64

x² = 64

x = +/- 8

*Speed cannot be negative, since it does not show direction. Therefore, we reject the x=-8 answer.

 

The speed of the current is 8mph.