Problem Statement: A man crosses a river in a boat. If he crosses the river in minimum time he takes 10 min with a drift 120 m. If he crosses the river taking shortest path, he takes 12.5 min, find (a) width of the river (b) velocity of the boat with respect to water (c) speed of the current? Explain in detail.

Solution:

Introduction: In this problem, we are given that a man crosses a river in a boat. The boat has a velocity with respect to water and the river has a current. We are given two scenarios, one in which the man crosses the river in minimum time and the other in which he crosses the river taking the shortest path. We need to find the width of the river, velocity of the boat with respect to water, and the speed of the current.

Scenario 1: Minimum Time

We are given that the man crosses the river in minimum time, which is 10 minutes. We are also given that the drift of the boat due to the current is 120 m.

Step 1: Find the distance covered by the boat

Distance covered by the boat is given by the formula:

Distance = Velocity x Time

Let the distance covered by the boat be D. Since the time taken by the man is 10 minutes, we can write:

D = V x 10

where V is the velocity of the boat with respect to water.

Step 2: Find the distance covered due to drift

Since the boat is drifting due to the current, it covers a distance of 120 m. Let us assume that the drift is in the y-direction.

Step 3: Find the width of the river

Let the width of the river be W. Using Pythagoras theorem, we can write:

D^2 = W^2 + 120^2

Substituting the value of D from step 1, we get:

100V^2 = W^2 + 120^2

Scenario 2: Shortest Path

We are given that the man crosses the river taking the shortest path, which takes 12.5 minutes.

Step 1: Find the distance covered by the boat

Let the distance covered by the boat be D. Since the time taken by the man is 12.5 minutes, we can write:

D = V x 12.5

where V is the velocity of the boat with respect to water.

Step 2: Find the distance covered due to current

Let the speed of the current be C. Since the boat is drifting due to the current, it covers a distance of C x 12.5 m. Let us assume that the drift is in the y-direction.

Step 3: Find the width of the river

Let the width of the river be W. Using Pythagoras theorem, we can write:

D^2 = W^2 + (C x 12.5)^2

Substituting the value of D from step 1, we get:

156.25V^2 = W^2 + (C x 12.5)^2

Step 4: Equate the two scenarios

Since