Points to Remember:
- This is a problem involving simultaneous equations.
- We need to define variables for the speed of the boat in still water and the speed of the stream.
- We will use the formula: time = distance / speed. Upstream speed is (boat speed – stream speed) and downstream speed is (boat speed + stream speed).
Introduction:
This question requires solving a system of simultaneous equations to determine the speed of a boat in still water and the speed of the stream. The problem utilizes the concept of relative speed, where the speed of the boat is affected by the current of the stream. Upstream travel is slower due to the opposing current, while downstream travel is faster due to the assisting current. We will use the given information about time and distance to create and solve the equations.
Body:
1. Defining Variables and Equations:
Let:
- ‘x’ represent the speed of the boat in still water (km/hr).
- ‘y’ represent the speed of the stream (km/hr).
Upstream speed = x – y
Downstream speed = x + y
We can translate the given information into two equations:
- Equation 1: (33/(x-y)) + (64/(x+y)) = 19 (Time taken for the first journey)
- Equation 2: (36/(x-y)) + (72/(x+y)) = 21 (Time taken for the second journey)
2. Solving the Simultaneous Equations:
To simplify, let’s substitute:
- a = 1/(x-y)
- b = 1/(x+y)
Our equations become:
- 33a + 64b = 19
- 36a + 72b = 21
We can solve this system using various methods, such as elimination or substitution. Let’s use elimination:
Multiply the first equation by 36 and the second equation by 33:
- 1188a + 2304b = 684
- 1188a + 2376b = 693
Subtract the first equation from the second:
72b = 9
b = 1/8
Substitute b = 1/8 into 33a + 64b = 19:
33a + 64(1/8) = 19
33a + 8 = 19
33a = 11
a = 1/3
Now substitute back to find x and y:
- a = 1/(x-y) = 1/3 => x – y = 3
- b = 1/(x+y) = 1/8 => x + y = 8
Adding the two equations: 2x = 11 => x = 5.5
Substituting x = 5.5 into x – y = 3: 5.5 – y = 3 => y = 2.5
3. Solution:
Therefore, the speed of the boat in still water (x) is 5.5 km/hr, and the speed of the stream (y) is 2.5 km/hr.
Conclusion:
By formulating and solving a system of simultaneous equations based on the given information about the boat’s upstream and downstream journeys, we have determined that the speed of the boat in still water is 5.5 km/hr and the speed of the stream is 2.5 km/hr. This problem highlights the practical application of algebraic techniques in solving real-world problems involving relative speeds. Understanding these concepts is crucial for navigation and other applications involving movement in fluids. Further research could explore the impact of varying stream speeds and different boat designs on travel times. A holistic approach to waterway management should consider factors like environmental sustainability and efficient transportation planning.
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