A Motorboat Travels 9 Miles Downstream (with The Current) In 30 Minutes. The Return Trip Upstream (against The Current) Takes 90 Minutes.Which System Of Equations Can Be Used To Find \[$x\$\], The Speed Of The Boat In Miles Per Hour, And

Feb 26, 2025 by ADMIN 238 views

A Motorboat's Speed: Solving the Downstream and Upstream Problem

Introduction

When a motorboat travels downstream, it is aided by the current, which increases its speed. Conversely, when it travels upstream, it is hindered by the current, which decreases its speed. In this article, we will explore how to use a system of equations to find the speed of the boat in miles per hour, denoted as x.

The Problem

A motorboat travels 9 miles downstream in 30 minutes. On the return trip, it takes 90 minutes to travel 9 miles upstream. We are asked to find the speed of the boat, x, in miles per hour.

Setting Up the Equations

Let's denote the speed of the boat in still water as x miles per hour and the speed of the current as c miles per hour. When the boat travels downstream, its speed is the sum of its speed in still water and the speed of the current, which is x + c miles per hour. Conversely, when it travels upstream, its speed is the difference between its speed in still water and the speed of the current, which is x - c miles per hour.

We can use the formula distance = rate × time to set up two equations based on the given information. For the downstream trip, we have:

9 = (x + c) × (30/60)

Simplifying the equation, we get:

9 = (x + c) × 0.5

For the upstream trip, we have:

9 = (x - c) × (90/60)

Simplifying the equation, we get:

9 = (x - c) × 1.5

Solving the System of Equations

We now have a system of two equations with two unknowns, x and c. We can solve this system using substitution or elimination. Let's use substitution.

Rearranging the first equation, we get:

x + c = 18

Rearranging the second equation, we get:

x - c = 6

Adding the two equations, we get:

2x = 24

Dividing both sides by 2, we get:

x = 12

Substituting x = 12 into one of the original equations, we get:

12 + c = 18

Subtracting 12 from both sides, we get:

c = 6

Conclusion

We have successfully solved the system of equations to find the speed of the boat, x, in miles per hour. The speed of the boat is 12 miles per hour, and the speed of the current is 6 miles per hour.

Discussion

This problem is a classic example of how to use a system of equations to solve a real-world problem. By setting up two equations based on the given information and solving the system using substitution or elimination, we can find the speed of the boat and the speed of the current.

Applications

This problem has many applications in real-world scenarios, such as:

Navigation: Knowing the speed of a boat and the speed of the current is crucial for navigation, especially in areas with strong currents.
Transportation: Understanding the speed of a boat and the speed of the current is essential for planning transportation routes and schedules.
Engineering: The speed of a boat and the speed of the current are critical factors in designing and building boats, bridges, and other infrastructure.

Future Work

In future work, we can explore more complex problems involving systems of equations, such as:

Multiple variables: We can add more variables to the system, such as the speed of the wind or the depth of the water.
Non-linear equations: We can use non-linear equations to model more complex relationships between variables.
Real-world applications: We can apply the concepts learned in this article to real-world problems in fields such as navigation, transportation, and engineering.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Glossary

System of equations: A set of two or more equations that involve two or more variables.
Substitution: A method of solving a system of equations by substituting one equation into another.
Elimination: A method of solving a system of equations by eliminating one variable by adding or subtracting the equations.
Rate: The speed at which an object moves, usually measured in miles per hour or kilometers per hour.
Time: The duration of a period, usually measured in hours, minutes, or seconds.

Introduction

In our previous article, we explored how to use a system of equations to find the speed of a motorboat in miles per hour. We set up two equations based on the given information and solved the system using substitution. In this article, we will answer some frequently asked questions about the problem and provide additional insights.

Q&A

Q: What is the speed of the boat in still water?

A: The speed of the boat in still water is 12 miles per hour.

Q: What is the speed of the current?

A: The speed of the current is 6 miles per hour.

Q: Why is the speed of the boat different downstream and upstream?

A: The speed of the boat is different downstream and upstream because the current affects its speed. When the boat travels downstream, the current aids its speed, making it faster. Conversely, when the boat travels upstream, the current hinders its speed, making it slower.

Q: How do you set up the equations for this problem?

A: To set up the equations, we use the formula distance = rate × time. We denote the speed of the boat in still water as x miles per hour and the speed of the current as c miles per hour. When the boat travels downstream, its speed is x + c miles per hour. Conversely, when it travels upstream, its speed is x - c miles per hour.

Q: What is the difference between the speed of the boat downstream and upstream?

A: The difference between the speed of the boat downstream and upstream is 12 - (-6) = 18 miles per hour.

Q: Can you use other methods to solve this problem?

A: Yes, we can use other methods to solve this problem, such as elimination or graphing. However, substitution is a straightforward and efficient method for solving this system of equations.

Q: What are some real-world applications of this problem?

A: This problem has many real-world applications, such as navigation, transportation, and engineering. Understanding the speed of a boat and the speed of the current is crucial for planning routes and schedules, designing boats and bridges, and ensuring safe passage.

Additional Insights

Speed and time: The speed of an object and the time it takes to travel a certain distance are inversely proportional. This means that if the speed of an object increases, the time it takes to travel a certain distance decreases, and vice versa.
Current and speed: The speed of a boat is affected by the current. When the boat travels downstream, the current aids its speed, making it faster. Conversely, when the boat travels upstream, the current hinders its speed, making it slower.
System of equations: This problem involves a system of two equations with two unknowns. We can solve this system using substitution, elimination, or graphing.

Conclusion

In this article, we answered some frequently asked questions about the problem and provided additional insights. We explored the speed of the boat in still water, the speed of the current, and the difference between the speed of the boat downstream and upstream. We also discussed real-world applications of this problem and provided additional information about speed and time, current and speed, and system of equations.