Boat Crossing the River
A boat has a velocity of \(8\,\text{km/h}\) in still water. Water flowing in a river has a velocity of \(6\,\text{km/h}\). The width of the river is \(4\sqrt{7}\,\text{km}\). The boat is steered so that it reaches directly opposite its starting point on the other bank.
What is the time taken by the boat to cross the river?
- (1) 2 hr
- (2) 2.5 hr
- (3) 3.2 hr
- (4) 4 hr
Select the Correct Option
Step‑by‑Step Explanation
Step 1: Determine the Required Component of the Boat’s Velocity
The boat’s speed in still water is \(8\,\text{km/h}\). In order to reach the point directly opposite on the other bank, the boat must steer in such a way that its upstream component cancels the river’s current (which flows at \(6\,\text{km/h}\)).
This means that the horizontal (along the river) component of the boat’s velocity must be \(6\,\text{km/h}\), leaving the remaining component for crossing the river.
Using the Pythagorean theorem, the component of the boat’s velocity across the river is: \[ v_{\text{across}} = \sqrt{(8\,\text{km/h})^2 - (6\,\text{km/h})^2} = \sqrt{64 - 36} = \sqrt{28} = 2\sqrt{7}\,\text{km/h}. \]
Step 2: Calculate the Time Taken to Cross the River
The width of the river is given as \(4\sqrt{7}\,\text{km}\). The time \(t\) to cross is:
\[
t = \frac{\text{Distance}}{\text{Speed}} = \frac{4\sqrt{7}\,\text{km}}{2\sqrt{7}\,\text{km/h}} = 2\,\text{hr}.
\]
Final Answer: The time taken by the boat to cross the river is \(2\,\text{hr}\), which corresponds to option (1).
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