A man can swim in still water at a speed of `5km//h`. Man crosses 1 km width of river along shortest possible path in 15 minutes . What is the speed of river in km/h?

Explanation

To solve the problem step by step, we will analyze the swimming motion of the man across the river and derive the speed of the river.

### Step 1: Understand the problem
The man swims across a river that is 1 km wide. He swims at a speed of 5 km/h in still water and takes 15 minutes to cross the river. We need to find the speed of the river.

### Step 2: Convert time from minutes to hours
The time taken to cross the river is given as 15 minutes. We need to convert this into hours for consistency with the speed units (km/h).

\[ \text{Time in hours} = \frac{15 \text{ minutes}}{60} = 0.25 \text{ hours} \]

### Step 3: Calculate the effective speed of the man across the river
The distance across the river is 1 km. The effective speed of the man across the river can be calculated using the formula:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

Substituting the values:

\[ \text{Effective speed} = \frac{1 \text{ km}}{0.25 \text{ hours}} = 4 \text{ km/h} \]

### Step 4: Set up the relationship between speeds
Let \( u \) be the speed of the river. The man swims at a speed of 5 km/h, and his effective speed across the river (which is perpendicular to the current) is given by the Pythagorean theorem:

\[ (5 \text{ km/h})^2 = (4 \text{ km/h})^2 + (u)^2 \]

### Step 5: Solve for the speed of the river
Substituting the values into the equation:

\[ 25 = 16 + u^2 \]

Now, rearranging the equation to solve for \( u^2 \):

\[ u^2 = 25 - 16 = 9 \]

Taking the square root:

\[ u = \sqrt{9} = 3 \text{ km/h} \]

### Conclusion
The speed of the river is \( 3 \text{ km/h} \).

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