Question

A motorboat traveling downstream covers the distance between port M and port N in 6 hours. Once, the motorboat stopped 40 km before reaching N, turned around, and returned to M. This took the motorboat 9 hours. Find the speed of the motorboat in still water if the speed of the current is 2 km/hour.

Answer 1

Answer:

18 km/h

Step-by-step explanation:

Let S km be the distance between ports M and N, x km/h be the speed of the motorboat in still water. Then x-2 km/h is the speed of the motorboat upstream and x+2 km/h is the speed of the motor boat downstream.

1. The motorboat traveling downstream covers the distance between port M and port N in 6 hours, then

$\dfrac{S}{x+2}=6$

2. Once, the motorboat stopped 40 km before reaching N, turned around, and returned to M. This took the motorboat 9 hours. Then

$\dfrac{S-40}{x+2}+\dfrac{S-40}{x-2}=9$

From the first equation $S=6(x+2)=6x+12.$ Substitute it into the second equation:

$\dfrac{6x+12-40}{x+2}+\dfrac{6x+12-40}{x-2}=9,\\ \\\dfrac{6x-28}{x+2}+\dfrac{6x-28}{x-2}=9.$

Now

$\dfrac{(6x-28)(x-2)+(6x-28)(x+2)}{(x-2)(x+2)}=9,\\ \\(6x-28)(x-2+x+2)=9(x^2-4),\\ \\(6x-28)\cdot 2x=9x^2-36,\\ \\12x^2-56x-9x^2+36=0,\\ \\3x^2-56x+36=0,\\ \\D=(-56)^2-4\cdot 3\cdot 36=2704,\\ \\x_{1,2}=\dfrac{56+\sqrt{2704}}{2\cdot 3}=\dfrac{56\pm52}{6}=\dfrac{2}{3},\ 18.$

The speed of the motorboat cannot be less than the speed of the current, thus, x=18 km/h.