A man who can swim 48 m/min in still water swims 200 m against the current and 200 m with the current. If the difference between those two times is 10 minutes, find the speed of the current.

Options

A. A)30 m/min
B. B)29 m/min
C. C)31m/min
D. D)32 m/min

Answer

To solve the problem, we need to find the speed of the current (let's denote it as \( K \)). We know that the man swims at a speed of 48 m/min in still water.

### Step-by-Step Solution:

1. **Define Variables**:
- Speed of the man in still water = 48 m/min
- Speed of the current = \( K \) m/min

2. **Calculate Effective Speeds**:
- When swimming against the current, the effective speed = \( 48 - K \) m/min
- When swimming with the current, the effective speed = \( 48 + K \) m/min

3. **Calculate Time Taken**:
- Time taken to swim 200 m against the current:
\[ \text{Time}_{\text{against}} = \frac{200}{48 - K} \]
- Time taken to swim 200 m with the current:
\[ \text{Time}_{\text{with}} = \frac{200}{48 + K} \]

4. **Set Up the Equation**:
- According to the problem, the difference in time between swimming against and with the current is 10 minutes:
\[ \frac{200}{48 - K} - \frac{200}{48 + K} = 10 \]

5. **Simplify the Equation**:
- Multiply through by the common denominator \((48 - K)(48 + K)\):
\[ 200(48 + K) - 200(48 - K) = 10(48 - K)(48 + K) \]
- This simplifies to:
\[ 200K + 200K = 10(48^2 - K^2) \]
- Combine like terms:
\[ 400K = 10(2304 - K^2) \]

6. **Rearrange the Equation**:
- Divide both sides by 10:
\[ 40K = 2304 - K^2 \]
- Rearranging gives:
\[ K^2 + 40K - 2304 = 0 \]

7. **Solve the Quadratic Equation**:
- Use the quadratic formula \( K = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 40, c = -2304 \):
\[ K = \frac{-40 \pm \sqrt{40^2 - 4 \cdot 1 \cdot (-2304)}}{2 \cdot 1} \]
- Calculate the discriminant:
\[ 1600 + 9216 = 10816 \]
- Now calculate \( K \):
\[ K = \frac{-40 \pm 104}{2} \]
- This gives two potential solutions:
\[ K = \frac{64}{2} = 32 \quad \text{(valid speed)} \]
\[ K = \frac{-144}{2} = -72 \quad \text{(not valid)} \]

8. **Conclusion**:
- The speed of the current \( K \) is 32 m/min.

### Final Answer:
The speed of the current is **32 m/min**.