Determining the Speed of a Boat in Calm Water

The speed of a boat in calm water is a crucial parameter for navigation and planning trips. This article discusses a method to determine the boat's speed in calm water by analyzing the distances and times of a boat's journey along and against a current. The variables and relationships between the distances, speeds, and times are analyzed through a system of equations.

Problem Statement

Consider a scenario where a boat travels 24 km downstream and 15 km upstream in 3 hours. In another trip, the boat travels 8 km downstream and 35 km upstream in 4 hours. We aim to find the speed of the boat in calm water. Let's denote:

Speed of the boat in calm water as b (in km/h)

Speed of the river current as r (in km/h)

Given Information

The two given trips are summarized below:

First Trip

Distance downstream: 24 km

Distance upstream: 15 km

Total time: 3 hours

Second Trip

Distance downstream: 8 km

Distance upstream: 35 km

Total time: 4 hours

Equations

For the first trip:

Effective speed downstream: ( b r )

Effective speed upstream: ( b - r )

The time taken is given by:

[ frac{24}{b r} frac{15}{b - r} 3 ]

For the second trip:

Effective speed downstream: ( b r )

Effective speed upstream: ( b - r )

The time taken is given by:

[ frac{8}{b r} frac{35}{b - r} 4 ]

Solving the Equations

To solve the equations, we begin with the first equation:

[ frac{24}{b r} frac{15}{b - r} 3 ]

Multiplying through by ((b r)(b - r)) gives:

[ 24(b - r) 15(b r) 3(b r)(b - r) ]

Expanding and simplifying:

[ 24b - 24r 15b 15r 3(b^2 - r^2) ]

[[ 39b - 9r 3b^2 - 3r^2 ]]

Combining like terms:

[ 3b^2 - 9b - 9r 3r^2 0 ]

Dividing by 3:

[ b^2 - 3b - 3r r^2 0 ]

Similarly, for the second equation:

[ frac{8}{b r} frac{35}{b - r} 4 ]

Multiplying through by ((b r)(b - r)) gives:

[ 8(b - r) 35(b r) 4(b r)(b - r) ]

Expanding and simplifying:

[ 8b - 8r 35b 35r 4(b^2 - r^2) ]

[[ 43b 27r 4b^2 - 4r^2 ]]

Combining like terms:

[ 4b^2 - 43b - 4r 4r^2 0 ]

We now have two quadratic equations:

[ b^2 - 3b - 3r r^2 0 ]

[ 4b^2 - 43b - 4r 4r^2 0 ]

Solving Simultaneously

Isolating (r) from the first equation:

[ r frac{3b^2 - 3b r^2}{3b - 3} ]

Substituting into the second equation:

[ 4b^2 - 43b - 4left(frac{3b^2 - 3b r^2}{3b - 3}right) - 4r 4r^2 0 ]

This results in a complex equation, so numerical methods or graphing is used to find the values of (b) and (r).

Numerical Solution

Using numerical methods or graphing calculators, we find the values of (b) and (r):

Approximately:

Speed of the boat in calm water (b approx 10) km/h

Speed of the river current (r approx 5) km/h

Conclusion

The speed of the boat in calm water is approximately 10 km/h. This value is crucial for navigation and planning river trips effectively.