The current of a stream runs at the rate of 4km an hour . A boat goes 6 km and comes back to the starting point in 2 hours .The speed of the boat in still water is

Options

A. 6 km/ hr
B. 8 km/hr
C. 7.5 km/hr
D. 6.8 km/hr

Answer

To solve the problem of finding the speed of a boat in still water given the current of a stream and the time taken for a round trip, we can follow these steps:

### Step 1: Define Variables
Let the speed of the boat in still water be \( x \) km/h. The speed of the stream is given as 4 km/h.

### Step 2: Determine Effective Speeds
When the boat is moving downstream (going with the current), its effective speed is:
\[ \text{Speed downstream} = x + 4 \text{ km/h} \]
When the boat is moving upstream (against the current), its effective speed is:
\[ \text{Speed upstream} = x - 4 \text{ km/h} \]

### Step 3: Calculate Time for Each Leg of the Journey
The distance for each leg of the journey is 6 km. The time taken to go downstream is:
\[ \text{Time downstream} = \frac{6}{x + 4} \]
The time taken to come back upstream is:
\[ \text{Time upstream} = \frac{6}{x - 4} \]

### Step 4: Set Up the Equation
According to the problem, the total time for the round trip is 2 hours. Therefore, we can set up the equation:
\[ \frac{6}{x + 4} + \frac{6}{x - 4} = 2 \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can multiply through by the common denominator, which is \((x + 4)(x - 4)\):
\[ 6(x - 4) + 6(x + 4) = 2(x + 4)(x - 4) \]

### Step 6: Expand and Simplify
Expanding both sides gives:
\[ 6x - 24 + 6x + 24 = 2(x^2 - 16) \]
This simplifies to:
\[ 12x = 2x^2 - 32 \]

### Step 7: Rearrange into Standard Form
Rearranging the equation gives:
\[ 2x^2 - 12x - 32 = 0 \]
Dividing through by 2 simplifies it to:
\[ x^2 - 6x - 16 = 0 \]

### Step 8: Factor the Quadratic Equation
To factor the quadratic, we look for two numbers that multiply to -16 and add to -6. The numbers -8 and 2 work:
\[ (x - 8)(x + 2) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives:
\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
Since speed cannot be negative, we discard \( x = -2 \).

### Step 10: Conclusion
Thus, the speed of the boat in still water is:
\[ \boxed{8} \text{ km/h} \]