A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?

1 Answer
Feb 3, 2018

The boat's speed is #10.4# km/h and it takes 1.8 hrs to travel upstream.

Explanation:

We know that #s = d/t#, thus #t = d/s#.

Therefore,

#d/("speed against current") + d/("speed with current") = t_"total"#

We know that #t_"total" = 3 hrs#, and that #d = 15# km. If we let the speed in still water be #x#, then the speed against the current would be #x - 2# and with the current #x + 2#. Therefore:

#3 = 15/(x - 2) + 15/(x + 2)#

#3(x^2 - 4) = 15(x + 2) + 15(x - 2)#

#3x^2 - 12 = 15x + 30 + 15x - 30#

#3x^2 - 30x - 12 = 0#

#x^2 - 10x - 4 = 0#

This cannot be factored, thus we use the quadratic formula or a graphical approach. Use your graphing calculator to enter #y_1 = x^2 - 10x - 4# and #y_2 = 0#. Then use the intersect function to determine the intersection points.

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A negative answer is impossible, so we deduce that the speed in still water of the boat is approximately #10.4 "km"/h#. When the boat is going upstream, its speed will be reduced by #2# km/h, to #8.4# km/h. Therefore,

#t = 15/8.4 = 1.8 hrs#

Hopefully this helps!