A motorboat went downstream for 28 km and immediately returned. It took the boat twice time to make returned trip than to go. If the speed of the river flow were twice as high, the trip downstream and back would take 672 minutes. Find the speed of the boat in still water and the speed of the river flow.

Answer

To solve the problem step by step, let's denote:

- \( x \) = speed of the boat in still water (in km/h)
- \( y \) = speed of the river flow (in km/h)

### Step 1: Establish the time equations
When the boat goes downstream, its effective speed is \( x + y \), and when it returns upstream, its effective speed is \( x - y \). The distance for both trips is 28 km.

The time taken to go downstream is given by:
\[ \text{Time downstream} = \frac{28}{x + y} \]

The time taken to return upstream is given by:
\[ \text{Time upstream} = \frac{28}{x - y} \]

According to the problem, the time taken for the return trip is twice that of the downstream trip:
\[ \frac{28}{x - y} = 2 \cdot \frac{28}{x + y} \]

### Step 2: Simplify the equation
We can simplify the equation by multiplying both sides by \( (x - y)(x + y) \):
\[ 28(x + y) = 56(x - y) \]

Expanding both sides gives:
\[ 28x + 28y = 56x - 56y \]

### Step 3: Rearranging the equation
Rearranging the terms to isolate \( x \) and \( y \):
\[ 28y + 56y = 56x - 28x \]
\[ 84y = 28x \]
Dividing both sides by 28:
\[ 3y = x \]

### Step 4: Analyze the second condition
The problem states that if the speed of the river flow were twice as high, the trip downstream and back would take 672 minutes. Thus, if \( y \) is doubled, the new speed of the river flow becomes \( 2y \).

The new effective speeds would be:
- Downstream: \( x + 2y \)
- Upstream: \( x - 2y \)

The total time for the round trip with the new river speed is:
\[ \text{Total time} = \frac{28}{x + 2y} + \frac{28}{x - 2y} = 672 \text{ minutes} = 11.2 \text{ hours} \]

### Step 5: Set up the equation for the new speeds
Setting up the equation:
\[ \frac{28}{x + 2y} + \frac{28}{x - 2y} = 11.2 \]

Multiplying through by \( (x + 2y)(x - 2y) \):
\[ 28(x - 2y) + 28(x + 2y) = 11.2(x^2 - 4y^2) \]
\[ 28x - 56y + 28x + 56y = 11.2x^2 - 44.8y^2 \]
\[ 56x = 11.2x^2 - 44.8y^2 \]

### Step 6: Substitute \( x = 3y \)
Substituting \( x = 3y \) into the equation:
\[ 56(3y) = 11.2(3y)^2 - 44.8y^2 \]
\[ 168y = 11.2(9y^2) - 44.8y^2 \]
\[ 168y = 100.8y^2 - 44.8y^2 \]
\[ 168y = 56y^2 \]
Dividing both sides by \( y \) (assuming \( y \neq 0 \)):
\[ 56y = 168 \]
\[ y = 3 \]

### Step 7: Find \( x \)
Now substituting \( y = 3 \) back into \( x = 3y \):
\[ x = 3 \times 3 = 9 \]

### Final Answer
The speed of the boat in still water is \( 9 \) km/h, and the speed of the river flow is \( 3 \) km/h.