The current in a stream moves at a speed of 10 km divided by hkm/h. A boat travels 19 km upstream and 19 km downstream in a total time of 19 hr.

Question

The speed of the boat in still water is ___ km/h

Answer 1

Hi Sarah!

 

With these kinds of questions, it's useful to gather the distance, rate, and time for both trips in this problem. Since the problem is asking for speed, the rate of the boat = x. Keep in mind that distance = rate * time

 

Upstream

Distance: 19 km

Rate: x km/h - 10 km/h (If the current pushes in the opposite direction subtract the current's speed from the boat's.)

Time: 19/(x-10) hrs      (because time = distance/rate)

 

Downstream

Distance: 19 km

Rate: x km/h + 10 km/h (This time you add because the current is making the boat go faster)

Time: 10/(x+10) hrs

 

We know that the total time is 19 hours, so the time upstream plus the time downstream has to equal 19. Therefore:

 

10/(x-10) + 10/(x+10) = 19

 

Find the common denominator: (x-10)(x+10)

Now you can make the denominators equal and add the numbers together.

 

19(x+10)/(x-10)(x+10) + 19(x-10)/(x-10)(x+10) = 19

 

(19x+190)/(x^2 -100) + (19x-190)/(x^2 - 100) = 19

 

19x+190+19x-190 = 19(x^2 - 100)

 

38x = 19x^2 -1900

 

0 = 19x^2 - 38x - 1900

 

Use your quadratic formula to get x = -9.050 and x = 11.050

 

Since we know the rate can't be negative if the boat was moving forward, we have to pick the positive one.

The boat's rate on still water is 11.050 km/h

 

Let me know if you have questions! :)