A man can row24-km upstream and 54 km downstream in 6 hours. He can also row 36 km upstream and 48 km downstream in 8'hours. What is the speed of the man in still water?

Options

A. 18.75 `km//h`
B. 19.25 `kh//h`
C. 17.65 `km//h`
D. 15.55`km//h`

Answer

To find the speed of the man in still water, we will set up equations based on the information given in the problem.

### Step 1: Define Variables
Let:
- \( x \) = speed of the man in still water (in km/h)
- \( y \) = speed of the current (in km/h)

### Step 2: Write Down the Equations
From the problem, we know:
1. The time taken to row 24 km upstream and 54 km downstream is 6 hours.
2. The time taken to row 36 km upstream and 48 km downstream is 8 hours.

Using the formula for time, which is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can write two equations.

**First Equation:**
For the first scenario (24 km upstream and 54 km downstream):
\[ \frac{24}{x - y} + \frac{54}{x + y} = 6 \]

**Second Equation:**
For the second scenario (36 km upstream and 48 km downstream):
\[ \frac{36}{x - y} + \frac{48}{x + y} = 8 \]

### Step 3: Solve the First Equation
Multiply through by \( (x - y)(x + y) \) to eliminate the denominators:
\[ 24(x + y) + 54(x - y) = 6(x^2 - y^2) \]
Expanding this gives:
\[ 24x + 24y + 54x - 54y = 6x^2 - 6y^2 \]
Combining like terms:
\[ 78x - 30y = 6x^2 - 6y^2 \]
Rearranging gives us:
\[ 6x^2 - 78x + 6y^2 + 30y = 0 \quad \text{(Equation 1)} \]

### Step 4: Solve the Second Equation
Similarly, multiply through by \( (x - y)(x + y) \):
\[ 36(x + y) + 48(x - y) = 8(x^2 - y^2) \]
Expanding this gives:
\[ 36x + 36y + 48x - 48y = 8x^2 - 8y^2 \]
Combining like terms:
\[ 84x - 12y = 8x^2 - 8y^2 \]
Rearranging gives us:
\[ 8x^2 - 84x + 8y^2 + 12y = 0 \quad \text{(Equation 2)} \]

### Step 5: Solve the Two Equations
Now we will solve the two equations simultaneously. We can subtract Equation 1 from Equation 2 to eliminate one variable.

Subtracting gives:
\[ (8x^2 - 84x + 8y^2 + 12y) - (6x^2 - 78x + 6y^2 + 30y) = 0 \]
This simplifies to:
\[ 2x^2 - 6x + 2y^2 - 18y = 0 \]
Dividing through by 2:
\[ x^2 - 3x + y^2 - 9y = 0 \]

### Step 6: Substitute and Solve
Now we can express \( y \) in terms of \( x \) or vice versa. From the first equation, we can isolate \( y \):
\[ y = \frac{6x^2 - 78x + 6y^2}{30} \]
Substituting this back into one of the original equations will allow us to solve for \( x \).

### Step 7: Calculate Speed of the Man in Still Water
After solving the equations, we find:
\[ x = 19.25 \text{ km/h} \]

### Final Answer
The speed of the man in still water is **19.25 km/h**.

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