Question

туристы на лодке гребли один час по течению реки и 30 минут пляли по течению,сложив весла.Затем они три часа гребли вверх по реке и прибыли к месту старта.Через сколько часов с момента старта вернулись бы туристы,если бы после часовой гребли по течеинию они сразу стали грести обратно?скорость лодки при гребле в стоячей воде и скорость течения реки постоянны

Answer 1

Problem Analysis

We are given that tourists rowed downstream for one hour and swam downstream for 30 minutes by folding their oars. Then, they rowed upstream for three hours and arrived at the starting point. We need to determine how many hours it would take for the tourists to return to the starting point if they started rowing back immediately after the one-hour row downstream. The speed of the boat and the speed of the river current are constant.

Solution

Let's assume the speed of the boat in still water is B and the speed of the river current is C.

When rowing downstream, the effective speed of the boat is the sum of the boat's speed and the river current's speed: B + C. The time spent rowing downstream is 1 hour, so the distance covered is (B + C) * 1.

When swimming downstream, the tourists are not using the boat's speed, so the effective speed is just the river current's speed: C. The time spent swimming downstream is 30 minutes, so the distance covered is C * (1/2).

When rowing upstream, the effective speed of the boat is the difference between the boat's speed and the river current's speed: B - C. The time spent rowing upstream is 3 hours, so the distance covered is (B - C) * 3.

Since the tourists arrived back at the starting point, the total distance covered rowing downstream, swimming downstream, and rowing upstream must be equal to the distance covered rowing downstream initially.

Using this information, we can set up the following equation:

(B + C) * 1 + C * (1/2) + (B - C) * 3 = (B + C) * x

where x is the number of hours it takes for the tourists to return to the starting point.

Simplifying the equation:

B + C + C/2 + 3B - 3C = Bx + Cx

4B - 2C = (B + C)x

4B - 2C = Bx + Cx

4B - Bx = 2C + Cx

Factoring out B and C:

B(4 - x) = C(2 + x)

Dividing both sides by (4 - x)(2 + x):

B = C(2 + x) / (4 - x)

We can solve this equation to find the relationship between the boat's speed and the river current's speed.

Let's substitute the given values into the equation and solve for B:

B = C(2 + x) / (4 - x)

Given that the speed of the boat in still water is equal to the speed of the river current, we can set B = C:

C = C(2 + x) / (4 - x)

Cross-multiplying:

C(4 - x) = C(2 + x)

Simplifying:

4 - x = 2 + x

2x = 2

x = 1

Therefore, it would take the tourists 1 hour to return to the starting point if they started rowing back immediately after the one-hour row downstream.

Answer

If the tourists started rowing back immediately after the one-hour row downstream, they would return to the starting point in 1 hour.

Note: The solution assumes that the speed of the boat in still water is equal to the speed of the river current.