find the speed of the boat going upstream and downstream

Question

a boat takes 6.5 hours to make a 70 mile trip upstream and 5 hours on the 70 mile return trip
. Let v be the speed of the boat in still water, and c be the speed of the current. the upstream speed of the boat is v-c and the downstream boat is v + c.

a) write 2 equations one for upstream part of the trip and one for the downstream  part, relating boat speed distance and time.

b)   solve the equations in part a for the speed of the current. round your answer to the nearest tenth of a mile per hour and show your work.

c)  how long would it take the boat to travel the 70 miles if there were no current?  round your answer to the nearest minute and show your work

Answer 1

First, recall that speed (or velocity) is a measure of the time required for an object to travel a certain distance. The speed of an object is given by the following formula:

          speed = distance/time     ==>     s = d/t

Letting v represent the speed of the boat in still water and c represent the speed of the current, we are given the following:

          upstream speed of boat =      s_upstream = v - c

      downstream speed of boat =  s_downstream = v + c

Upstream data:     d = 70 miles ,   t = 6.5 hours

     s_upstream = 70 miles/6.5 hours = 70/6.5 (miles/hr) = 10.8 miles/hr              

Downstream data:     d = 70 miles ,   t = 5 hours

      s_downstream = 70 miles/5 hours = 10/5 (miles/hr) = 14 miles/hr

a)

With this, we arrive at the following equations for the upstream and downstream speeds of the boat:

        s_upstream:       v - c = 10.8 

     s_downstream:     v + c = 14     

b)

To solve for the speed of the current (c), first solve each equation for the speed of the boat in still water (v) in terms of c. Then we can set these to expressions equal to one another and solve for c. That is,

        s_upstream:      v - c = 10.8     ==>     v = 10.8 + c

     s_downstream:     v + c = 14        ==>     v = 14 - c

                           10.8 + c = 14 - c

          10.8 - 10.8 + c + c = 14 - 10.8 - c + c

                                    2c = 3.2

                                2c/2 = 3.2/2

                                     c = 1.6

Thus, the speed of the current (c) is 1.6 miles/hr.

c)

Notice that if we add the two equations in part a to one another, c (the speed of the current) will cancel out and leave us with one variable to solve for, that being the speed of the boat in still water (v):

             v - c = 10.8

     +     v + c = 14

   ________________________

     v - c + v + c = 10.8 + 14

                     2v = 24.8

                 2v/2 = 24.8/2

                      v = 12.4

Thus, the speed of the boat in still water (i.e., with no current) is 12.4 miles/hr.

Answer 2

Hi Sabrina,

Let's take each part of this multipart problem one step at a time and all will be clear. 

Part (a):

We can set up 2 equations using the distance formula:  D = RT

Our formula tells us that the distance (D) is the product of the rate or speeed (R) and the time (T).

Using our given info, we can use our distance formula to set up our 2 equations:

1.  Our upstream given info is D = 70, R = v - c (speed going against the current), T = 6.5; so our upstream equation is 6.5(v - c) = 70

2.  Our downstream given info is D = 70, R = v + c (speed going with the current), T = 5.0; so our downstream equation is 5.0(v + c) = 70

Part (b):

We can solve our two equations for c by the elimination technique (combining the two equations and eliminating the variable v):

Equation 1:  6.5(v - c) = 70 (equation from part a)

Equation 2:  6.5v - 6.5c = 70 (multiplication results for left side of Equation 1)

Equation 3:  5.0(v + c) = 70 (equation from part a)

Equation 4:  5v + 5c =70 (multiplication results for left side of Equation 3)

Equation 5:  32.5v - 32.5c = 350 (multiplying both sides of Equation 2 by 5 to set up elimination of v)

Equation 6:  32.5v + 32.5c = 455 (multiplying both sides of equation 4 by 6.5 to set up elimination of v)

Equation 7:  65c = 105 (subtracting Equation 6 from equation 5 to eliminate v)

Dividing both sides of Equation 7 by 65, we get c = 1.6 mph (speed of current rounded to nearest tenth)

Part (c):

We can use either equation from part (a) to calculate speed of boat in still water by plugging in our answer of 1.6 for c (speed of current) obtained in part (b):

Upstream Equation:

6.5(v - c) = 70 (from part a)

6.5(v - 1.6) = 70 (plugging in answer for c from part b)

6.5v - 10.4 = 70 (multiplication results for left side of equation)

6.5v = 80.4 (adding 10.4 to both sides of equation to isolate variable term on left side)

Dividing both sides of equation by 6.5, we get v = 12.4 mph (velocity of boat in still water)

Downstream Equation:

5.0(v + c) = 70 (from part a)

5.0(v + 1.6) = 70 (plugging in answer for c from part b)

5v + 8 = 70 (multiplication results for left side of equation)

5v = 62 (subtracting 8 from both sides of equation to isolate variable term on left side)

Dividing both sides of equation by 5, we get v = 12.4 mph (velocity of boat in still water)

So we can see that our result for speeed of boat in stiil water (v = 12.4 mph) is the same using either equation from part (a) and plugging in our answer for c (speed of current) obtained in part (b), which is what we would expect.

I trust the above will prove helpful to your understanding of this problem.

Thanks for submitting your question.

Regards, Jordan.