Please Answer!!!

Question

A boat traveled 210 miles downstream and back. The trip downstream took10 hours. The trip back took 70 hours. What is the speed of the boat still in water? What is the speed of the current?

Answer 1

Tom,

 

Let's define the variables for each of our unknowns. We know the distance and we know time, but we don't know the rate (speed) of the boat or the current. So let's us:

   

     b = speed of the boat (mph)

     c = speed of the current (mph)

 

If you are going downstream, are you going with or against the current? Well as it took 10 hours versus 70 hours, you must be going faster downstream, thus having the current working to speed you up. So downstream you have the speed of the boat (b) plus the speed of the current (c), so your combined speed is b + c.

  

    downstream speed = b + c (mph)

 

Leaving you to have to work against the current when boating upstream, or standing outside in the downpours today! And if the current is going against you, it will be slowing you down, or reducing your speed. And taking it 7 times longer to go the same distance. So upstream you have the speed of the boat (b) less the speed of the current (c), so your combined speed is b - c. 

 

    upstream speed = b - c (mph)

 

So now we have:

 

                              Downstream          Upstream

   Distance (miles)        210                     210

   Rate (mph)              b + c                   b - c

   Time (hours)              10                      70

 

Not set up your equations: d=rt

 

   Downstream:  210 = 10(b + c)  or  b + c = 21  (divide both sides by 10)

   Upstream:      210 = 70(b - c)     or  b - c = 3  (divide both sides by 70)

 

Now you have a system of equations to solve: (you can choose to solve for b or c and substitute or to use elimination and add the two equations. I'll use elimination.)

 

     b + c = 21

 +  b - c =  3

    2b     = 24

 

      b = 12                    Substitute b = 12 into 210 = 10(b + c)

                                                                    210 = 10(12 + c)

                                                                    210 = 120 + 10c

                                                                     90 = 10c

                                                                       9 = c                        Check: 210 = 70(b - c)

                                                                                                                  210 = 70(12 - 9)

                                                                                                                  210 = 70(3)

                                                                                                                  210 = 210   √

 

So your boats speed is 12 miles per hour and the currents speed is 9 miles per hour.

 

Answer 2

Tom,

 

The total trip is 210 miles, which includes going down once and coming back once. So the downstream trip and upstream trip are both 105 miles.

 

There are two unknowns: Boat speed in MPH (B) and current speed in MPH (C). Two unknowns, so you need two equations.

 

Downstream trip

 

105 miles in 10 hours. The boat is going with the current. So

 

105 miles = B*10 hours + C*10 hours

 

Upstream trip

 

105 miles in 70 hours. The boat is going against the current. So

 

105 miles = B*70 hours - C*70 hours

 

Two equations and two variables. Use substitution and/or linear combinations to find B & C. Leave a comment if you'd like more help!

 

Bryce

 

Answer 3

This is a system of equations problem. 

The boats rate with the current on the way downstream is 210miles/10hours or 21mph

Its rate against the current on the way back is 210miles/70hours or 3mph

 

So is b=speed of boat in still water and c is speed of the current

 

b+c=21

b-c=3

 

if you solve using elimination you add the two equations together and the c term cancels leaving

 

2b=24

divide both sides by 2

 

b=12mph

 

plug this back into one of the equations for b

 

12+c=21

subtract 12 from both sides

c=9mph

 

you can check your work by plugging both terms into both equations

 

12+9=21

12-9=3