A river 4 miles wide is flowing at the rate of2 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?

To determine the minimum time taken by a boat to cross a river that is 4 miles wide, while the river flows at a rate of 2 miles per hour and the boat has a speed of 4 miles per hour in still water, we can break this problem down into manageable parts.

Understanding the Scenario

We have a river that is 4 miles wide, flowing at 2 miles per hour. The boat can move at a speed of 4 miles per hour in still water. However, since the river is flowing, the boat's effective speed will be influenced by the current when it attempts to cross.

Visualizing the Problem

Imagine the river as a rectangle where the width is 4 miles. The boat starts on one bank and needs to reach the opposite bank. The river's current will push the boat downstream as it crosses. To minimize the time taken, the boat should head slightly upstream to counteract the current.

Calculating the Effective Speed

The boat's speed relative to the ground is a combination of its speed across the river and the speed of the current. If the boat heads directly across the river, it will be carried downstream by the current. To find the angle at which the boat should head, we can use some trigonometry.

• Let the angle the boat makes with the perpendicular to the riverbank be θ.
• The component of the boat's speed across the river is \(4 \cos(θ)\).
• The component of the boat's speed against the current is \(4 \sin(θ)\).

To counteract the current, the upstream component of the boat's speed must equal the speed of the river:

4 sin(θ) = 2

From this, we can solve for sin(θ):

sin(θ) = 0.5

Thus, θ = 30 degrees.

Finding the Time to Cross

Now that we know the angle, we can find the effective speed of the boat across the river:

Effective speed across the river = 4 cos(30 degrees) = 4 × (√3/2) = 2√3 miles per hour.

Next, we can calculate the time taken to cross the river:

Time = Distance / Speed

Time = 4 miles / (2√3 miles per hour) = 4 / (2√3) = 2/√3 hours.

Converting Time to a More Usable Format

To express this in a more familiar format, we can convert hours into minutes:

2/√3 hours ≈ 1.155 hours, which is approximately 69.3 minutes.

Final Thoughts

In summary, the minimum time taken by the boat to cross the river is approximately 1.155 hours or about 69.3 minutes. This approach not only illustrates the importance of vector components in navigation but also highlights how to effectively counteract currents when crossing a river.