Sandra's Shortest Path Solving A Mathematical Word Problem

In this article, we will solve a classic mathematical word problem involving distance and direction. This type of problem often appears in various fields, from basic geometry to navigation and physics. We will break down the problem step-by-step, apply relevant mathematical concepts, and arrive at a solution that tells us the shortest path Sandra could take to her friend's house. Understanding how to solve such problems enhances our analytical and problem-solving skills, which are essential in numerous real-world scenarios. Whether you are a student looking to improve your math skills or someone interested in the practical application of mathematics, this article will provide a clear and concise solution to the word problem.

Problem Statement

The word problem states that Sandra initially walked 100 yards in a direction 45° north of east. Following this, she turned and walked an additional 400 yards straight east to reach her friend's house. The challenge is to determine the direct distance and direction Sandra would need to walk to reach her friend's house if she were to take the shortest possible route. This requires us to combine vector addition and trigonometry to find the resultant displacement. Let's delve into the steps to solve this problem.

Visualizing the Problem

To effectively solve this problem, it's crucial to visualize Sandra's journey. Imagine a coordinate plane where Sandra starts at the origin (0,0). Her first walk is 100 yards at a 45° angle north of east. This can be represented as a vector with both horizontal (eastward) and vertical (northward) components. Her second walk is 400 yards straight east, which can be represented as a horizontal vector. The shortest path is a straight line from her starting point to her friend's house, which is the resultant vector of the two walks. By visualizing this, we can better understand the components and apply the correct mathematical principles to find the solution.

Breaking Down the First Leg of the Journey

Sandra's first leg involves walking 100 yards at a 45° angle north of east. To find the components of this vector, we use trigonometry. The eastward component is given by 100 * cos(45°), and the northward component is given by 100 * sin(45°). Since cos(45°) and sin(45°) are both approximately 0.707, the eastward component is approximately 100 * 0.707 = 70.7 yards, and the northward component is also approximately 100 * 0.707 = 70.7 yards. These components are crucial for adding the vectors and finding the resultant direction and distance. Understanding these calculations is essential for solving similar problems involving vectors and angles.

Analyzing the Second Leg of the Journey

Sandra's second leg is straightforward: she walks 400 yards straight east. This movement can be represented as a vector with a magnitude of 400 yards in the eastward direction and no northward component. This leg is simpler to analyze because it only involves movement along one axis. To find the total eastward displacement, we will add this 400 yards to the eastward component of her first leg. Understanding this leg is vital for calculating the total displacement and finding the direct path to her friend's house. The combination of both legs will give us the necessary information to determine the shortest route.

Calculating the Total Eastward Displacement

To find the total eastward displacement, we add the eastward components of both legs of Sandra's journey. The first leg has an eastward component of approximately 70.7 yards, and the second leg has an eastward displacement of 400 yards. Therefore, the total eastward displacement is 70.7 + 400 = 470.7 yards. This value represents the total horizontal distance Sandra traveled east from her starting point. Understanding how to calculate this component is essential for determining the magnitude and direction of the resultant displacement. The total eastward displacement is a key part of finding the shortest path.

Determining the Total Northward Displacement

The total northward displacement is determined by the northward component of Sandra's first leg, as the second leg involves only eastward movement. The northward component of the first leg is approximately 70.7 yards. Since there is no additional northward movement in the second leg, the total northward displacement remains 70.7 yards. This value represents the total vertical distance Sandra traveled north from her starting point. Knowing the total northward displacement, along with the total eastward displacement, allows us to calculate the direct distance and direction Sandra would need to walk to reach her friend's house. This step is crucial in solving the problem and finding the shortest path.

Finding the Resultant Distance

To find the resultant distance, which is the straight-line distance from Sandra's starting point to her friend's house, we use the Pythagorean theorem. The total eastward displacement (470.7 yards) and the total northward displacement (70.7 yards) form the two legs of a right triangle, and the resultant distance is the hypotenuse. Therefore, the resultant distance is the square root of (470.7^2 + 70.7^2). Calculating this gives us approximately √(221556.49 + 4998.49) = √226554.98 ≈ 476 yards. This is the shortest distance Sandra could have walked to reach her friend's house. Understanding the Pythagorean theorem and its application here is crucial for solving similar problems.

Calculating the Direction

To find the direction Sandra would need to walk, we use trigonometry, specifically the arctangent function. The direction is the angle θ north of east, where tan(θ) is the ratio of the northward displacement to the eastward displacement. Thus, θ = arctan(70.7 / 470.7). Calculating this gives us θ ≈ arctan(0.15) ≈ 8.53 degrees. Therefore, Sandra would need to walk approximately 476 yards in a direction 8.53° north of east to take the shortest path to her friend's house. This calculation completes the solution, providing both the distance and direction of the shortest path.

Final Answer

In conclusion, if Sandra wanted to walk the shortest distance to her friend's house, she would need to walk approximately 476 yards in a direction 8.53° north of east. This solution was achieved by breaking down the problem into components, using vector addition, and applying the Pythagorean theorem and trigonometric functions. This problem illustrates the practical application of mathematical concepts in real-world scenarios. Understanding these principles is invaluable for solving a variety of similar problems involving distance, direction, and vectors. The solution provides a clear and concise answer to the initial word problem, showcasing the power of mathematical analysis.