A person rows a distance of 20 km upstream and 15 km downstream in a total time of `2(5)/(12)` hours. If the speed of the current is 4 km/h, then in how many hours can be row a distance of 64 kın upstream?

Options

A. `5(1)/(3)`
B. `4(2)/(3)`
C. `5(1)/(2)`
D. `4(1)/(4)`

Answer

To solve the problem step by step, we will follow the given information and derive the necessary equations.

### 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 \( y = 4 \) km/h.

### Step 2: Determine Upstream and Downstream Speeds
- **Upstream Speed**: The speed of the boat when rowing upstream is \( x - y = x - 4 \) km/h.
- **Downstream Speed**: The speed of the boat when rowing downstream is \( x + y = x + 4 \) km/h.

### Step 3: Total Time Equation
The total time taken to row 20 km upstream and 15 km downstream is given as \( 2 \frac{5}{12} \) hours. We convert this mixed fraction into an improper fraction:
\[ 2 \frac{5}{12} = \frac{24 + 5}{12} = \frac{29}{12} \text{ hours} \]

### Step 4: Write the Time Equations
Using the formula for time (Time = Distance / Speed), we can write:
- Time taken to row upstream: \( \frac{20}{x - 4} \)
- Time taken to row downstream: \( \frac{15}{x + 4} \)

Setting up the equation for total time:
\[ \frac{20}{x - 4} + \frac{15}{x + 4} = \frac{29}{12} \]

### Step 5: Solve the Equation
To solve the equation, we first find a common denominator:
\[ \frac{20(x + 4) + 15(x - 4)}{(x - 4)(x + 4)} = \frac{29}{12} \]
Expanding the numerator:
\[ 20x + 80 + 15x - 60 = 35x + 20 \]
Thus, we have:
\[ \frac{35x + 20}{x^2 - 16} = \frac{29}{12} \]

Cross-multiplying gives:
\[ 12(35x + 20) = 29(x^2 - 16) \]
Expanding both sides:
\[ 420x + 240 = 29x^2 - 464 \]
Rearranging the equation:
\[ 29x^2 - 420x - 704 = 0 \]

### Step 6: Apply the Quadratic Formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 29, b = -420, c = -704 \):
\[ x = \frac{420 \pm \sqrt{(-420)^2 - 4 \cdot 29 \cdot (-704)}}{2 \cdot 29} \]
Calculating the discriminant:
\[ (-420)^2 = 176400, \quad 4 \cdot 29 \cdot 704 = 81664 \]
Thus:
\[ x = \frac{420 \pm \sqrt{176400 + 81664}}{58} \]
\[ x = \frac{420 \pm \sqrt{258064}}{58} \]
Calculating the square root:
\[ \sqrt{258064} = 508 \]
So:
\[ x = \frac{420 + 508}{58} = \frac{928}{58} = 16 \text{ km/h} \]

### Step 7: Calculate Upstream Speed
Now, we find the upstream speed:
\[ \text{Upstream Speed} = x - 4 = 16 - 4 = 12 \text{ km/h} \]

### Step 8: Calculate Time to Row 64 km Upstream
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{64}{12} = \frac{16}{3} \text{ hours} = 5 \frac{1}{3} \text{ hours} \]

### Final Answer
The time taken to row 64 km upstream is \( 5 \frac{1}{3} \) hours.

---