A boat takes 4 hours to travel from a place X to Y downstream and back from Yto X upstream.If the distance from X to Y is 10.5 km, and the speed ofthe current is 9 km/h, then the speed of the boat in still water, in km/h is:

Options

A. `10(1)/(2)`
B. 15
C. 12
D. `12(1)/(2)`

Answer

To solve the problem step by step, we will use the information provided to set up equations based on the speeds of the boat and the current.

### Step 1: Define Variables
Let the speed of the boat in still water be \( X \) km/h. The speed of the current is given as \( 9 \) km/h.

### Step 2: Determine Downstream and Upstream Speeds
- When the boat is moving downstream (from X to Y), the effective speed of the boat is:
\[ \text{Speed downstream} = X + 9 \text{ km/h} \]
- When the boat is moving upstream (from Y to X), the effective speed of the boat is:
\[ \text{Speed upstream} = X - 9 \text{ km/h} \]

### Step 3: Calculate Time Taken for Each Journey
The distance from X to Y is \( 10.5 \) km. The time taken to travel downstream and upstream can be expressed as:
- Time taken downstream:
\[ \text{Time downstream} = \frac{10.5}{X + 9} \]
- Time taken upstream:
\[ \text{Time upstream} = \frac{10.5}{X - 9} \]

### Step 4: Set Up the Total Time Equation
According to the problem, the total time for the round trip (downstream and upstream) is \( 4 \) hours. Therefore, we can write the equation:
\[ \frac{10.5}{X + 9} + \frac{10.5}{X - 9} = 4 \]

### Step 5: Simplify the Equation
To simplify, we can multiply through by the common denominator, which is \((X + 9)(X - 9)\):
\[ 10.5(X - 9) + 10.5(X + 9) = 4(X + 9)(X - 9) \]
This simplifies to:
\[ 10.5X - 94.5 + 10.5X + 94.5 = 4(X^2 - 81) \]
Combining like terms gives:
\[ 21X = 4X^2 - 324 \]

### Step 6: Rearrange into Standard Quadratic Form
Rearranging the equation results in:
\[ 4X^2 - 21X - 324 = 0 \]

### Step 7: Solve the Quadratic Equation
We can use the quadratic formula \( X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = -21 \), and \( c = -324 \):
\[ b^2 - 4ac = (-21)^2 - 4 \cdot 4 \cdot (-324) = 441 + 5184 = 5625 \]
Now, taking the square root:
\[ \sqrt{5625} = 75 \]
Now substituting back into the quadratic formula:
\[ X = \frac{21 \pm 75}{8} \]
Calculating the two potential solutions:
1. \( X = \frac{96}{8} = 12 \)
2. \( X = \frac{-54}{8} \) (not valid as speed cannot be negative)

Thus, the speed of the boat in still water is:
\[ X = 12 \text{ km/h} \]

### Final Answer
The speed of the boat in still water is \( 12 \) km/h.

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