A boat sails downstream from a point P to point Q. which is 24 km away from P and then returns to P. If the actual speed of the boat (in still water) is 6 km/h. the entire trip from P to Q takes 3 hours less than that from Q to P. What should be the actual speed (in km/h) of the boat to sail from P to Q in 2 hours?

Options

A. 12
B. 15
C. 10
D. 22

Answer

To solve the problem step by step, we need to determine the speed of the boat in still water that allows it to sail from point P to point Q in 2 hours.

### Step 1: Define the variables
Let:
- \( x \) = speed of the stream (in km/h)
- The speed of the boat in still water = 6 km/h

### Step 2: Determine the downstream and upstream speeds
- Downstream speed (from P to Q) = speed of the boat + speed of the stream = \( 6 + x \) km/h
- Upstream speed (from Q to P) = speed of the boat - speed of the stream = \( 6 - x \) km/h

### Step 3: Set up the time equations
The distance from P to Q is 24 km. The time taken to travel downstream from P to Q is given by:
\[ \text{Time}_{\text{downstream}} = \frac{24}{6 + x} \]
The time taken to travel upstream from Q to P is given by:
\[ \text{Time}_{\text{upstream}} = \frac{24}{6 - x} \]

According to the problem, the entire trip from P to Q takes 3 hours less than that from Q to P:
\[ \frac{24}{6 + x} + 3 = \frac{24}{6 - x} \]

### Step 4: Solve the equation
Rearranging the equation gives:
\[ \frac{24}{6 - x} - \frac{24}{6 + x} = 3 \]
To eliminate the fractions, we can multiply through by \((6 - x)(6 + x)\):
\[ 24(6 + x) - 24(6 - x) = 3(6 - x)(6 + x) \]
This simplifies to:
\[ 24x + 24x = 3(36 - x^2) \]
\[ 48x = 108 - 3x^2 \]
Rearranging gives:
\[ 3x^2 + 48x - 108 = 0 \]
Dividing the entire equation by 3:
\[ x^2 + 16x - 36 = 0 \]

### Step 5: Factor or use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 16, c = -36 \):
\[ x = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 1 \cdot (-36)}}{2 \cdot 1} \]
\[ x = \frac{-16 \pm \sqrt{256 + 144}}{2} \]
\[ x = \frac{-16 \pm \sqrt{400}}{2} \]
\[ x = \frac{-16 \pm 20}{2} \]
Calculating the two possible values:
1. \( x = \frac{4}{2} = 2 \) (valid)
2. \( x = \frac{-36}{2} = -18 \) (not valid)

Thus, \( x = 2 \) km/h.

### Step 6: Calculate the speed to sail from P to Q in 2 hours
To sail from P to Q in 2 hours, we need the downstream speed to be:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{24}{2} = 12 \text{ km/h} \]
Since the downstream speed is \( 6 + x \):
\[ 6 + x = 12 \]
Substituting \( x = 2 \):
\[ 6 + 2 = 8 \text{ km/h} \]

### Final Answer
The actual speed of the boat to sail from P to Q in 2 hours is:
\[ \text{Speed in still water} = 12 - 2 = 10 \text{ km/h} \]