Calculating Speed of a Boat in Still Water Using Distance and Time Data

The problem of determining the speed of a boat in still water based on its downstream and upstream travel data is a classic application of basic algebra and physics principles. This article will guide you through solving such problems using real examples and provide insights on the underlying formulas.

Problem Formulation

Consider a boat traveling downstream a distance of 40 kilometers and upstream a distance of 24 kilometers in a total of 4 hours, given that the current of the river is 4 km/hr.

Step-by-Step Solution

Let's denote the speed of the boat in still water as B (km/hr). The speed of the river current is specified as 4 km/hr.

When traveling downstream: The effective speed of the boat will be B 4 km/hr.

When traveling upstream: The effective speed of the boat will be B - 4 km/hr.

We know that time distance/speed. Therefore, the total time taken for the trip can be expressed using the following equation:

$$frac{40}{B 4} frac{24}{B - 4} 4$$

This equation represents the total time spent traveling downstream and upstream, which is 4 hours.

Solving the Equation

To solve the equation, we will first find a common denominator and simplify:

begin{align*} frac{40}{B 4} frac{24}{B - 4} 4 40(B - 4) 24(B 4) 4(B 4)(B - 4) 40B - 160 24B 96 4(B^2 - 16) 64B - 64 4B^2 - 64 64B 4B^2 16B B^2 B^2 - 16B 0 B(B - 16) 0end{align*}

This results in two solutions: B 0 or B 16 km/hr. Since a speed of 0 doesn't make sense in this context, the valid solution is:

$$B 16 , text{km/hr}.$$

Verification

Let's verify the solution:

Downstream speed: (16 4 20 , text{km/hr})

Upstream speed: (16 - 4 12 , text{km/hr})

$$frac{40}{20} frac{24}{12} 2 2 4 , text{hours}$$

The solution checks out, confirming that the speed of the boat in still water is indeed 16 km/hr.

Additional Examples

Here are a few more examples to illustrate different methods of solving similar problems:

Example 1

A boat travels 36 km in downstream and 24 km in upstream, taking 4 hours in total. The current of the river is 4 km/hr.

Example 2

A boat travels 32 km in 4 hours downstream and 24 km in 6 hours upstream. The current of the river is unknown.

Example 3

A boat travels 25 km in 2 hours downstream and in 5 hours upstream. The current of the river is unknown.

Conclusion

By understanding and applying the principles outlined in these examples, one can solve a variety of problems related to boat speed calculations. Whether you're an enthusiast, a student, or a professional, these techniques will be invaluable for tackling real-world scenarios involving boat travel in flowing water.