The width of a rivers is 25m and in it water is flowing with a velocity of `4m//min` . A boatman is standing on the bank of the river . He want to sail the boat to a point at the other bank which is directly opposite to him . In what time will he cross the river , if he can sail the boat at `8m//min`. relative to the water ?

Explanation

To solve the problem step by step, we need to analyze the situation involving the boatman, the river, and the velocities involved.

### Step 1: Understand the Given Information
- Width of the river (d) = 25 m
- Velocity of the river (V_r) = 4 m/min
- Velocity of the boat relative to the water (V_b) = 8 m/min

### Step 2: Determine the Velocity of the Boat Across the River
The boatman wants to reach a point directly opposite him on the other bank. To achieve this, he needs to counteract the downstream flow of the river. The effective velocity of the boat across the river (V_b_perpendicular) can be calculated using the Pythagorean theorem because the boat's velocity can be broken down into two components: one perpendicular to the river (across) and one parallel to the river (downstream).

Using the formula:
\[ V_b^2 = V_b_{\text{perpendicular}}^2 + V_r^2 \]

We can rearrange this to find the perpendicular component:
\[ V_b_{\text{perpendicular}} = \sqrt{V_b^2 - V_r^2} \]

Substituting the values:
\[ V_b_{\text{perpendicular}} = \sqrt{(8 \, \text{m/min})^2 - (4 \, \text{m/min})^2} \]
\[ V_b_{\text{perpendicular}} = \sqrt{64 - 16} \]
\[ V_b_{\text{perpendicular}} = \sqrt{48} \]
\[ V_b_{\text{perpendicular}} = 4\sqrt{3} \, \text{m/min} \]

### Step 3: Calculate the Time to Cross the River
Now that we have the effective velocity of the boat across the river, we can calculate the time (t) it takes to cross the river using the formula:
\[ t = \frac{d}{V_b_{\text{perpendicular}}} \]

Substituting the values:
\[ t = \frac{25 \, \text{m}}{4\sqrt{3} \, \text{m/min}} \]

To simplify:
\[ t = \frac{25}{4\sqrt{3}} \]

To calculate this, we can rationalize the denominator:
\[ t = \frac{25\sqrt{3}}{12} \]

Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ t \approx \frac{25 \times 1.732}{12} \]
\[ t \approx \frac{43.3}{12} \]
\[ t \approx 3.61 \, \text{minutes} \]

### Final Answer
The time it will take for the boatman to cross the river is approximately **3.61 minutes**.

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