A river 4.0 miles wide is following at the rate of 2 miles/hr. The minimum time taken by a boat to cross the river with a speed v=4 miles/hr (in still water) is approximately

Explanation

To solve the problem of finding the minimum time taken by a boat to cross a river that is 4.0 miles wide while the river flows at a rate of 2 miles/hr, and the boat has a speed of 4 miles/hr in still water, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Width of the river (D) = 4.0 miles
- Speed of the river (Vr) = 2 miles/hr
- Speed of the boat in still water (Vb) = 4 miles/hr

2. **Determine the Effective Velocity of the Boat:**
The boat's effective velocity when crossing the river is a combination of its speed in still water and the speed of the river. To find the resultant velocity (Vbg) of the boat with respect to the ground, we can use the Pythagorean theorem since the boat's direction is perpendicular to the river's flow.

\[ Vbg = \sqrt{Vb^2 + Vr^2} \]
Substituting the values:
\[ Vbg = \sqrt{(4 \text{ miles/hr})^2 + (2 \text{ miles/hr})^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \text{ miles/hr} \]

3. **Calculate the Time to Cross the River:**
The time (T) taken to cross the river can be calculated using the formula:
\[ T = \frac{D}{Vbg} \]
Substituting the known values:
\[ T = \frac{4 \text{ miles}}{2\sqrt{5} \text{ miles/hr}} = \frac{4}{2\sqrt{5}} = \frac{2}{\sqrt{5}} \text{ hours} \]

4. **Approximate the Time:**
To get a numerical approximation, we can calculate:
\[ \sqrt{5} \approx 2.236 \Rightarrow \frac{2}{\sqrt{5}} \approx \frac{2}{2.236} \approx 0.894 \text{ hours} \]
Converting this to minutes:
\[ 0.894 \text{ hours} \times 60 \text{ minutes/hour} \approx 53.64 \text{ minutes} \]

### Final Answer:
The minimum time taken by the boat to cross the river is approximately **0.894 hours** or about **53.64 minutes**.

---