Question

Лодка проплыла 18 км по течению реки и 6 км против течения, затратив на весь путь 4ч. Чему равна собственная скорость лодки, если скорость течения равна 3 км /ч?

Answer 1

Problem Analysis

We are given that a boat traveled 18 km downstream and 6 km upstream in a total of 4 hours. The speed of the river current is given as 3 km/h. We need to find the speed of the boat in still water.

Solution

Let's assume the speed of the boat in still water is x km/h.

When the boat is traveling downstream, its effective speed is the sum of its own speed and the speed of the current. So, the effective speed downstream is (x + 3) km/h.

When the boat is traveling upstream, its effective speed is the difference between its own speed and the speed of the current. So, the effective speed upstream is (x - 3) km/h.

We can use the formula distance = speed × time to calculate the time taken for each leg of the journey.

The time taken to travel downstream is given by: 18 km = (x + 3) km/h × t1 hours The time taken to travel upstream is given by: 6 km = (x - 3) km/h × t2 hours We are also given that the total time for the entire journey is 4 hours: t1 + t2 = 4 hours We can solve this system of equations to find the value of x, the speed of the boat in still water.

Solution Steps

1. From equation we can solve for t1: t1 = 18 km / (x + 3) km/h

2. From equation we can solve for t2: t2 = 6 km / (x - 3) km/h

3. Substitute the values of t1 and t2 into equation 18 km / (x + 3) km/h + 6 km / (x - 3) km/h = 4 hours

4. Simplify the equation by multiplying through by the common denominator: 18(x - 3) + 6(x + 3) = 4(x + 3)(x - 3)

5. Expand and simplify the equation: 18x - 54 + 6x + 18 = 4(x^2 - 9)

6. Simplify further: 24x - 36 = 4x^2 - 36

7. Rearrange the equation to form a quadratic equation: 4x^2 - 24x = 0

8. Factor out 4x: 4x(x - 6) = 0

9. Solve for x: x = 0 or x = 6

Since the speed of the boat cannot be zero, the speed of the boat in still water is 6 km/h.

Answer

The speed of the boat in still water is 6 km/h.