Deducing Paddling Speed in Still Water: A Mathematical Approach

Mang Kevin is a vendor who lives 7 km upstream from the town. When the current is 4 km/hr, he can paddle his boat downstream to the town and come back in 2 hours. What is the average paddling speed in still water? This article will explore the mathematical calculations needed to determine the average speed of his boat in still water.

Exploring the Problem

The problem requires us to find the speed at which Mang Kevin can paddle his boat in still water (without the influence of the current). This involves understanding the time taken to travel upstream and downstream, accounting for the influence of the current.

Solution

Firstly, letapos;s define the variables:

S: The speed of the boat in still water (in km/hr)

4: The speed of the current (in km/hr)

7: The distance traveled upstream (or downstream, as the distance doesnapos;t change)

When the boat travels downstream, it moves with the current, so its effective speed is (S 4). Conversely, when it travels upstream, it moves against the current, making its effective speed (S - 4).

Time Calculation

The time taken to travel 7 km downstream is ( frac{7}{S 4} ) hours, and the time taken to travel 7 km upstream is ( frac{7}{S - 4} ) hours. Since the total time for the round trip is 2 hours, we can form the equation:

[frac{7}{S 4} frac{7}{S - 4} 2]

This equation can be simplified and solved using algebraic manipulation to determine the value of S.

Simplifying the Equation

First, multiply both sides of the equation by the common denominator ((S 4)(S - 4)) to clear the fractions:

[frac{7(S - 4) 7(S 4)}{(S 4)(S - 4)} 2]

Simplifying the numerator:

[frac{7S - 28 7S 28}{S^2 - 16} 2]

Which reduces to:

[frac{14S}{S^2 - 16} 2]

By cross-multiplying, we get:

[14S 2(S^2 - 16)]

Distributing the 2 on the right side:

[14S 2S^2 - 32]

Bringing all terms to one side of the equation:

[2S^2 - 14S - 32 0]

Dividing the entire equation by 2 to simplify:

[S^2 - 7S - 16 0]

Solving the Quadratic Equation

This quadratic equation can be solved using the quadratic formula, which is given by:

[S frac{-b pm sqrt{b^2 - 4ac}}{2a}]

For the equation (S^2 - 7S - 16 0), the coefficients are:

- (a 1)- (b -7)- (c -16)

Substituting these values into the quadratic formula:

[S frac{-(-7) pm sqrt{(-7)^2 - 4(1)(-16)}}{2(1)}]

Which simplifies to:

[S frac{7 pm sqrt{49 64}}{2}]

This further simplifies to:

[S frac{7 pm sqrt{113}}{2}]

Since (sqrt{113} approx 10.63), the solutions are:

[S frac{7 pm 10.63}{2}]

This gives us two solutions:

[S frac{7 10.63}{2} approx 8.815] and [S frac{7 - 10.63}{2} approx -1.815]

The negative value is not physically meaningful in this context, so the speed of the boat in still water is approximately 8.815 km/hr.

Conclusion

The average paddling speed in still water is approximately 8.815 km/hr. This calculation takes into account the influence of the current and ensures that the total round-trip time matches the given value of 2 hours. Understanding this concept is crucial for real-world scenarios involving boats and streams, optimizing travel time and distance.

For more details and to learn how to apply similar mathematical techniques to solve other problems, consider consulting a physics or mathematics textbook or practicing more problems online.