The boat covered 15 km upstream and 6 km downstream, spending the same amount of time as it would

The boat covered 15 km upstream and 6 km downstream, spending the same amount of time as it would have if it had walked 22 km on the lake. What is the boat’s own speed if the speed of the river is 2 km / h.

Let the boat’s own speed be x km / h, then the speed of the boat along the river is (x + 2) km / h, and the speed of the boat against the river is (x – 2) km / h. The boat covered 6 km along the river in 6 / (x + 2) hours, and 15 km against the river in 15 / (x – 2) hours, and 22 km along the lake in 22 / x hours. According to the condition of the problem, it is known that the boat spent (6 / (x + 2) + 15 / (x – 2)) hours on the way along the river and against the river, or as much as on the movement along the lake, i.e. 22 / x hours. Let’s make an equation and solve it.

6 / (x + 2) + 15 / (x – 2) = 22 / x;

O.D.Z. x ≠ ± 2; x ≠ 0;

(6x (x – 2) + 15x (x + 2)) / (x (x ^ 2 – 4)) = (22 (x ^ 2 – 4)) / (x (x ^ 2 – 4));

6x (x – 2) + 15x (x + 2) = 22 (x ^ 2 – 4);

6x ^ 2 – 12x + 15x ^ 2 + 30x = 22x ^ 2 – 88;

21x ^ 2 + 18x – 22x ^ 2 + 88 = 0;

-x ^ 2 + 18x + 88 = 0;

x ^ 2 – 18x – 88 = 0;

D = (-18) ^ 2 – 4 * 1 * (-88) = 324 + 352 = 676; √D = 26;

x = (-b ± √D) / (2a);

x1 = (18 + 26) / 2 = 44/2 = 22 (km / h);

x2 = (18 – 26) / 2 = -8/2 = -4 – the speed cannot be negative.

Answer. 22 km / h