On a river, Q is the midpoint between two points P and R on the same bank. A boat takes 16 hours 40 minutes to go from P to R. It takes 12 hours to go from P to Q and return. How long will it take to go from R to P?

Points to Remember:

This is a problem involving the speed of a boat in still water and the speed of the river current.
We need to use the concept of relative speed to solve this problem.
The time taken to travel upstream is different from the time taken to travel downstream.

Introduction:

This question is a classic example of a problem involving relative speed. It tests our understanding of how the speed of a boat is affected by the current of a river. When a boat travels downstream (with the current), its effective speed is the sum of its speed in still water and the speed of the current. Conversely, when traveling upstream (against the current), its effective speed is the difference between its speed in still water and the speed of the current. We will use this principle to solve the problem.

Body:

1. Defining Variables:

Let:

v_b = speed of the boat in still water
v_r = speed of the river current
d = distance between P and R (therefore, distance between P and Q is d/2)

2. Setting up Equations:

P to R (downstream): Time = 16 hours 40 minutes = 16 + (40/60) = 16.67 hours. The distance is d. Therefore, d = (v_b + v_r) * 16.67
P to Q and back (upstream and downstream): Time = 12 hours. Distance P to Q is d/2.
- Downstream (P to Q): Time = (d/2) / (v_b + v_r)
- Upstream (Q to P): Time = (d/2) / (v_b – v_r)
  Total time: (d/2) / (v_b + v_r) + (d/2) / (v_b – v_r) = 12

3. Solving the Equations:

We have two equations with three unknowns. However, we are only interested in the time taken to go from R to P (upstream). Let’s simplify the second equation:

(d/2) * [(v_b – v_r) + (v_b + v_r)] / [(v_b + v_r)(v_b – v_r)] = 12
d * v_b / (v_bÂ² – v_rÂ²) = 12

From the first equation, d = (v_b + v_r) * 16.67. Substituting this into the simplified second equation:

[(v_b + v_r) * 16.67 * v_b] / (v_bÂ² – v_rÂ²) = 12

This equation is still difficult to solve directly for v_b and v_r. However, we can observe that the time taken from R to P (upstream) will be:

Time (R to P) = d / (v_b – v_r)

Since d = (v_b + v_r) * 16.67, we can substitute:

Time (R to P) = [(v_b + v_r) * 16.67] / (v_b – v_r)

Without further information or simplification, we cannot find a numerical solution for the time taken from R to P. More information is needed to solve for individual speeds (v_b and v_r).

Conclusion:

The problem highlights the complexities of relative motion. While we can set up equations based on the given information, we cannot definitively determine the time taken to travel from R to P without additional data, such as the speed of the boat in still water or the speed of the river current. To solve this type of problem completely, we need at least one more independent equation relating the boat’s speed and the river’s speed. Further information is required to provide a numerical answer. The approach outlined above demonstrates the correct method for tackling such problems involving relative speeds. Future problems of this type should include sufficient information to allow for a complete solution.

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