Solving Motorboat Speed With Systems Of Equations
Hey guys! Let's dive into a classic word problem involving a motorboat, currents, and the magic of systems of equations. We're going to break down how to find the speed of a boat in still water and the speed of the current itself. This problem is a great example of how math can help us understand and solve real-world scenarios. So, buckle up, and let's get started!
Understanding the Problem: Downstream and Upstream
The core of this problem revolves around the concept of a motorboat traveling both downstream (with the current) and upstream (against the current). This difference in direction significantly impacts the boat's overall speed. When a boat travels downstream, the current aids its progress, effectively increasing its speed. Think of it as a helpful push. Conversely, when the boat travels upstream, the current hinders its progress, acting as a resistance that slows it down. This is similar to walking on a moving walkway – you move faster with the walkway, and slower against it.
Here’s the setup we're working with:
- The motorboat travels 9 miles downstream in 30 minutes.
- The return trip upstream takes 90 minutes.
Our mission is to figure out two key things:
x: The speed of the boat in still water (miles per hour).y: The speed of the current (miles per hour).
The key to solving this is to translate the problem into mathematical equations. We'll use the fundamental relationship: distance = speed x time or, rearranged, speed = distance / time.
Now, let's think about how the current affects the boat's speed. When the boat goes downstream, the current's speed is added to the boat's speed. When the boat goes upstream, the current's speed is subtracted from the boat's speed.
So, downstream speed = x + y and upstream speed = x - y.
Before we can formulate our equations, we must make sure all our units are consistent. Since we want our answer in miles per hour, we must convert the time from minutes to hours.
- 30 minutes = 0.5 hours
- 90 minutes = 1.5 hours
Now we have all the pieces we need to get to work!
Formulating the Equations: The Math Behind the Journey
Alright, let's translate the problem into two equations. Remember, the speed of the boat is affected by the current which impacts how fast the boat moves relative to the ground. We have the distance and time for both downstream and upstream journeys, and we can find the speed using the formula: speed = distance / time. Let's break it down step-by-step to make sure everyone understands!
Downstream:
- Distance = 9 miles
- Time = 0.5 hours
- Downstream speed =
x + y
So, the downstream speed is 9 miles / 0.5 hours = 18 mph. This gives us our first equation:
x + y = 18
This equation says the boat's speed in still water (x) plus the current's speed (y) equals 18 mph. That's the combined speed while traveling downstream.
Upstream:
- Distance = 9 miles
- Time = 1.5 hours
- Upstream speed =
x - y
So, the upstream speed is 9 miles / 1.5 hours = 6 mph. This gives us our second equation:
x - y = 6
This equation tells us that the boat's speed in still water (x) minus the current's speed (y) equals 6 mph. This is the boat's speed while fighting against the current.
Now, we have a system of equations:
x + y = 18x - y = 6
These two equations represent the problem in a way that allows us to find the unknowns, x and y. This is the fundamental setup for solving this type of word problem. We're now set up to solve for x and y. This system of equations is the key to unlocking the speeds we need.
Let’s move on to actually solving for x and y, and find out the speed of the boat in still water and the speed of the current!
Solving the System of Equations: Finding the Speeds
Okay, team! Now that we have our system of equations, it's time to solve for x (boat speed) and y (current speed). There are several methods we can use, such as substitution, elimination, or graphing. For this example, let's use the elimination method because it's particularly well-suited to this type of problem.
The elimination method involves adding or subtracting the equations to eliminate one of the variables. Notice that we have +y in the first equation and -y in the second. This is perfect for elimination!
Let's add the two equations together:
(x + y) + (x - y) = 18 + 6
Simplifying, we get:
2x = 24
Now, divide both sides by 2 to solve for x:
x = 12
So, the speed of the boat in still water (x) is 12 mph! Awesome! We've solved for one of our unknowns. We're almost there!
Now, let's substitute the value of x (12) into either of the original equations to solve for y. Let's use the first equation: x + y = 18.
Substitute 12 for x:
12 + y = 18
Subtract 12 from both sides:
y = 6
Therefore, the speed of the current (y) is 6 mph.
We did it! We’ve solved the problem and found both the boat's speed and the current's speed. These equations have allowed us to find the answers! This system of equations gave us all the information we needed.
The Answer: Final Thoughts
So, to recap, the motorboat's speed in still water is 12 mph, and the speed of the current is 6 mph. Now we can write out our final answer to the problem:
x(boat speed) = 12 mphy(current speed) = 6 mph
This means that the boat is capable of traveling at 12 mph, but the current either helps or hinders it depending on the direction of travel. This is a very common type of problem, and knowing how to solve it can be quite helpful. Understanding the concept of downstream and upstream travel, combined with the ability to set up and solve a system of equations, gives us the power to conquer a wide variety of similar problems. We took the problem and turned it into the tools necessary to solve it!
I hope you guys found this breakdown useful! Systems of equations might seem tricky at first, but with practice, they become a powerful tool for solving all sorts of real-world problems. Keep practicing, and you'll be a pro in no time! Remember the core ideas and practice, and you'll become more and more proficient at solving similar problems. Keep on learning, and don't be afraid to ask questions. Happy math-ing! Remember to revisit these concepts as they'll surely appear again in future math adventures. Keep up the great work! And that's all, folks!