Calculating Distance A Step By Step Solution For Samantha And Mia's Journey
Introduction
In this article, we will delve into a classic problem involving distance, rate, and time. The scenario presents Samantha and Mia leaving Julia's house simultaneously, traveling in different directions – Mia walking north and Samantha running west. Our goal is to determine the distance between them after one hour, rounding the answer to the nearest tenth. This problem is a great example of applying the Pythagorean theorem in a real-world context. We'll break down the problem step-by-step, ensuring a clear understanding of the concepts involved. This article is designed for students, math enthusiasts, and anyone interested in problem-solving techniques in mathematics. Understanding these concepts can be valuable in various fields, from navigation to physics. By the end of this guide, you'll not only know the answer to this specific problem but also gain a deeper understanding of how to approach similar problems involving distance, rate, and time. We will explore the underlying principles and demonstrate how to apply them effectively. So, let's embark on this mathematical journey and unravel the solution together. Remember, the key to mastering math is not just memorizing formulas but understanding the logic behind them. We'll focus on building that understanding throughout this article. Let's dive in and explore the fascinating world of distance calculations!
Problem Statement
The problem states that Samantha and Mia each left Julia's house at the same time. Mia walked north at a rate of 7 kilometers per hour, while Samantha ran west at 11 kilometers per hour. We need to find the distance between them after one hour. This is a classic problem that combines concepts of speed, time, and distance, and introduces the application of the Pythagorean theorem. The problem implicitly assumes that both Samantha and Mia maintain a constant speed throughout the hour. It also assumes that they are traveling in perfectly perpendicular directions – Mia directly north and Samantha directly west. These assumptions are crucial for us to apply the Pythagorean theorem, which applies specifically to right-angled triangles. To solve this problem effectively, we need to break it down into smaller, manageable steps. First, we'll calculate the individual distances traveled by Mia and Samantha. Then, we'll use these distances to form a right-angled triangle, where the distance between them is the hypotenuse. Finally, we'll apply the Pythagorean theorem to find the length of the hypotenuse, which will give us the final answer. This step-by-step approach will not only help us solve this problem but also provide a framework for tackling similar problems in the future. Remember, problem-solving in mathematics is a skill that develops with practice and understanding of fundamental concepts. So, let's proceed with the solution, keeping these principles in mind.
Solution: Calculating Individual Distances
To begin, we need to calculate the distance each person traveled in one hour. Recall the fundamental formula: Distance = Rate × Time. Mia walked north at 7 kilometers per hour. Since she walked for one hour, the distance she covered is:
Distance (Mia) = 7 km/hour × 1 hour = 7 kilometers.
Similarly, Samantha ran west at 11 kilometers per hour. In one hour, she covered:
Distance (Samantha) = 11 km/hour × 1 hour = 11 kilometers.
Now that we know the distances each person traveled, we can visualize their paths as two sides of a right-angled triangle. Mia's northward journey forms one side, and Samantha's westward journey forms the other. The distance between them is the hypotenuse of this triangle. This is a crucial step in understanding the problem, as it allows us to apply the Pythagorean theorem. Visualizing the problem in this way makes the solution more intuitive. It's also a good practice to draw a diagram to represent the situation. A simple sketch can often clarify the relationships between different quantities and make the problem easier to solve. In this case, a right-angled triangle with sides 7 km and 11 km clearly illustrates the problem. The next step is to apply the Pythagorean theorem to find the length of the hypotenuse, which represents the distance between Samantha and Mia after one hour. So, let's move on to the next section and see how the Pythagorean theorem helps us find the solution.
Applying the Pythagorean Theorem
The Pythagorean theorem states that in a right-angled 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. Mathematically, this is expressed as: a² + b² = c², where 'a' and 'b' are the lengths of the two shorter sides, and 'c' is the length of the hypotenuse. In our case, Mia's distance (7 kilometers) and Samantha's distance (11 kilometers) are the two shorter sides of the right-angled triangle. The distance between them is the hypotenuse, which we need to find. Let's denote the distance between Samantha and Mia as 'd'. Applying the Pythagorean theorem, we get:
7² + 11² = d²
Calculating the squares:
49 + 121 = d²
Adding the numbers:
170 = d²
To find 'd', we need to take the square root of both sides:
d = √170
Now, we need to calculate the square root of 170. Using a calculator, we find:
d ≈ 13.038 kilometers
The problem asks us to round the answer to the nearest tenth. So, rounding 13.038 to the nearest tenth gives us 13.0 kilometers. Therefore, the distance between Samantha and Mia after one hour is approximately 13.0 kilometers. This result highlights the power of the Pythagorean theorem in solving real-world problems. It allows us to relate distances traveled in perpendicular directions and find the straight-line distance between two points. Understanding and applying the Pythagorean theorem is a fundamental skill in geometry and has wide applications in various fields.
Final Answer and Conclusion
After applying the Pythagorean theorem and performing the necessary calculations, we have determined that the distance between Samantha and Mia after one hour is approximately 13.0 kilometers. This answer is rounded to the nearest tenth, as requested in the problem statement. To recap, we first calculated the individual distances traveled by Mia and Samantha using the formula Distance = Rate × Time. Then, we recognized that these distances form the two shorter sides of a right-angled triangle, with the distance between them being the hypotenuse. We applied the Pythagorean theorem (a² + b² = c²) to find the length of the hypotenuse, which gave us the distance between Samantha and Mia. This problem effectively demonstrates how mathematical concepts can be applied to solve real-world scenarios. It reinforces the importance of understanding fundamental formulas and theorems, such as the distance formula and the Pythagorean theorem. Moreover, it highlights the value of visualizing problems and breaking them down into smaller, manageable steps. Problem-solving in mathematics is not just about finding the right answer; it's also about developing critical thinking skills and understanding the underlying principles. By working through this problem, we have not only found the solution but also strengthened our understanding of these important mathematical concepts. Remember, practice is key to mastering these skills. So, keep exploring different problems and applying these techniques to enhance your problem-solving abilities.
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