Distance Traveled: Carlos, Ana, And Perpendicular Paths
Hey guys! Let's dive into a fun math problem involving Carlos and Ana, who left their house for work. This problem involves understanding distances, perpendicular paths, and a little bit of algebra. We'll break it down step by step so it's super easy to follow. So, grab your thinking caps, and let's get started!
Understanding the Problem
Okay, so here's the scenario: Carlos and Ana leave from the same point in their building's garage. After one minute, they've traveled in perpendicular directions. Think of it like they're making a right angle with their paths. We know that they are 13 meters apart at this point. The key piece of information here is that Carlos traveled 7 meters more than Ana. Our mission, should we choose to accept it, is to figure out how far Ana traveled.
This problem is a classic example of how math concepts can be applied to real-world situations. We're dealing with distances, relative speeds, and geometric relationships – all wrapped into one intriguing question. To solve this, we'll need to dust off our knowledge of the Pythagorean theorem and a bit of algebraic manipulation. Don't worry, it sounds scarier than it actually is! We'll walk through it together, making sure each step makes perfect sense. The beauty of this problem lies in its simplicity; once you visualize the scenario and understand the relationships, the solution unfolds quite naturally. So, let's get ready to visualize, calculate, and conquer this distance dilemma!
Visualizing the Scenario
Before we jump into calculations, let's paint a picture in our minds. Imagine the garage as a starting point, and Carlos and Ana moving away from it. Since they travel in perpendicular directions, their paths form a right angle. This is super important because it allows us to use the Pythagorean theorem, which, as you might remember, deals with the relationship between the sides of a right triangle. Now, picture a straight line connecting Carlos and Ana after that one minute. This line forms the hypotenuse of our right triangle, and its length is 13 meters (the distance between them).
Think of Ana's distance as one leg of the triangle, Carlos's distance as the other leg, and the 13-meter separation as the hypotenuse. The fact that Carlos traveled 7 meters more than Ana gives us another crucial piece of the puzzle. We can represent Ana's distance as 'x' and Carlos's distance as 'x + 7'. Now we have all the components we need to set up our equation using the Pythagorean theorem. Visualizing the problem in this way transforms it from an abstract word problem into a concrete geometric scenario. This makes the problem much more approachable and easier to solve. So, with this mental image in place, we're ready to move on to the next step: setting up the equation.
Setting Up the Equation
Alright, guys, now comes the fun part where we translate our visualized scenario into a mathematical equation. Remember the Pythagorean theorem? It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms: a² + b² = c², where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides.
In our case, Ana's distance is one side (let's call it 'x'), Carlos's distance is the other side ('x + 7'), and the distance between them (13 meters) is the hypotenuse. So, we can plug these values into the Pythagorean theorem. Our equation looks like this: x² + (x + 7)² = 13². See how we've transformed the word problem into a neat, solvable equation? This is a crucial step in problem-solving – taking the information and expressing it in a way that allows us to use mathematical tools. Now, the next step is to simplify this equation and solve for 'x'. This will involve expanding the squared term, combining like terms, and potentially using the quadratic formula. Don't worry, we'll take it one step at a time, making sure everything is clear and straightforward. So, let's roll up our sleeves and get ready to crunch some numbers!
Solving the Equation
Okay, let's tackle this equation! We've got x² + (x + 7)² = 13². The first thing we need to do is expand that (x + 7)² term. Remember, (x + 7)² means (x + 7) * (x + 7). Using the FOIL method (First, Outer, Inner, Last) or any method you prefer, we get x² + 14x + 49.
Now, let's substitute that back into our equation: x² + (x² + 14x + 49) = 13². Next, we need to square 13, which gives us 169. So, our equation now looks like this: x² + x² + 14x + 49 = 169. Let's simplify further by combining like terms. We have two x² terms, which combine to 2x². So, our equation becomes: 2x² + 14x + 49 = 169. Now, to solve this quadratic equation, we need to set it equal to zero. We can do this by subtracting 169 from both sides: 2x² + 14x + 49 - 169 = 0. This simplifies to: 2x² + 14x - 120 = 0. We can make things even simpler by dividing the entire equation by 2: x² + 7x - 60 = 0. Now we have a simplified quadratic equation that we can solve. We can use factoring, the quadratic formula, or completing the square. Factoring is often the quickest method if it's possible. So, let's see if we can factor this equation in the next section.
Factoring the Quadratic Equation
Now that we've simplified our equation to x² + 7x - 60 = 0, let's try factoring it. Factoring involves finding two numbers that multiply to -60 (the constant term) and add up to 7 (the coefficient of the x term). Think of pairs of factors of 60: 1 and 60, 2 and 30, 3 and 20, 4 and 15, 5 and 12, 6 and 10. We need a pair that has a difference of 7 since we have a -60. Aha! 5 and 12 fit the bill. To get a +7x, we need +12 and -5. So, we can rewrite our equation as (x + 12)(x - 5) = 0.
Now, using the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero, we can set each factor equal to zero: x + 12 = 0 or x - 5 = 0. Solving for x in each case, we get: x = -12 or x = 5. We have two possible solutions for x, but remember what x represents – Ana's distance traveled. Distance can't be negative, so we discard x = -12. This leaves us with x = 5 as the only valid solution. Therefore, Ana traveled 5 meters. But hold on, we're not quite done yet! We still need to answer the original question completely. We've found Ana's distance, but let's make sure we understand the whole picture in the next step.
Final Answer
Alright, we've cracked the code! We found that x = 5, which means Ana traveled 5 meters. Now, let's think about what the question asked and make sure we've fully answered it. The question was: "What distance did Ana travel?" We've done the math, and we've got our answer.
So, the final answer is: Ana traveled 5 meters. We started with a word problem, visualized the scenario, set up a Pythagorean equation, solved for the unknown, and now we have our solution. Isn't it satisfying when everything comes together like that? This problem demonstrates how math can be used to solve real-world scenarios. By understanding the relationships between distances and using tools like the Pythagorean theorem and factoring, we can find the answers we're looking for. Great job, guys! You've tackled this problem like pros. Remember, the key to solving word problems is to break them down into smaller, manageable steps. Visualize, set up the equation, solve, and then make sure you've answered the original question. Keep practicing, and you'll become math whizzes in no time!