Calculating Horizontal Distance Plank Against Wall Using Pythagorean Theorem

Introduction to the Plank Problem

In this article, we will delve into a classic problem involving a plank of wood leaning against a wall, and we'll use the Pythagorean Theorem to find a specific dimension. This is a common type of problem in mathematics that combines geometry and algebra, providing a practical application of a fundamental theorem. Imagine a scenario where you have a plank of wood, 32 inches long, leaning against a wall. The wall has a height of 15 inches where the plank touches it. Our goal is to calculate the horizontal distance between the base of the wall and the bottom of the plank. This problem not only tests our understanding of the Pythagorean Theorem but also highlights its relevance in real-world situations. Understanding these concepts is crucial for students and anyone interested in problem-solving using mathematical principles. We'll break down the problem step by step, ensuring a clear understanding of how to apply the theorem and arrive at the correct solution. The Pythagorean Theorem, which 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, is our key tool here. By identifying the sides of the right-angled triangle formed by the plank, wall, and ground, we can set up an equation and solve for the unknown distance. This exercise is not just about finding a numerical answer; it's about developing a logical approach to problem-solving and understanding how mathematical theorems can be applied in practical contexts.

Problem Statement: Plank, Wall, and Distance

To begin, let's clearly define the problem we aim to solve. We have a 32-inch plank of wood resting against a wall, reaching a height of 15 inches on the wall. The primary task is to determine the horizontal distance between the bottom of the wall and the point where the plank touches the ground. This scenario creates a right-angled triangle, with the plank acting as the hypotenuse, the wall as one leg, and the ground as the other leg. Visualizing this setup is crucial for applying the Pythagorean Theorem effectively. The problem requires us to use our knowledge of geometry and algebra to find the missing side of the triangle. It's a practical example of how mathematical concepts can be used to solve real-world problems. The challenge lies in correctly identifying the given values, setting up the equation based on the Pythagorean Theorem, and solving for the unknown variable. This process involves algebraic manipulation and careful calculation to arrive at the accurate answer, rounded to one decimal place as specified. By breaking down the problem into smaller, manageable steps, we can ensure a clear and logical approach to finding the solution. Understanding the problem statement thoroughly is the first step towards successfully applying the Pythagorean Theorem and determining the required horizontal distance. This problem-solving exercise enhances our understanding of spatial relationships and the practical applications of mathematical principles in everyday situations. The ability to visualize and interpret such scenarios is a valuable skill, particularly in fields like engineering, architecture, and physics.

Applying the Pythagorean Theorem

The Pythagorean Theorem is the cornerstone of our solution. It states that in a right-angled triangle, a² + b² = c², where c is the length of the hypotenuse (the side opposite the right angle), and a and b are the lengths of the other two sides. In our scenario, the plank (32 inches) acts as the hypotenuse, the wall (15 inches) is one side, and the horizontal distance we want to find is the other side. We can represent the horizontal distance as x. Therefore, we can set up the equation as follows: 15² + x² = 32². This equation is the key to solving the problem. It translates the geometric relationships into an algebraic expression that we can manipulate to find the value of x. The next step involves solving this equation for x, which requires us to perform algebraic operations such as squaring, subtracting, and taking the square root. By carefully following these steps, we can isolate x and determine the horizontal distance between the wall and the bottom of the plank. The Pythagorean Theorem provides a powerful tool for solving problems involving right-angled triangles, and its application here demonstrates its practical utility. Understanding how to apply this theorem is essential for anyone studying geometry and related fields. This problem not only reinforces the theorem itself but also highlights the importance of translating real-world scenarios into mathematical models. By mastering these skills, we can approach a wide range of problems with confidence and accuracy. The ability to apply the Pythagorean Theorem effectively is a fundamental skill in mathematics and has numerous applications in various fields.

Step-by-Step Solution

Now, let’s walk through the step-by-step solution to find the horizontal distance. We've established the equation 15² + x² = 32². The first step is to calculate the squares of the known values: 15² equals 225, and 32² equals 1024. So, our equation becomes 225 + x² = 1024. Next, we need to isolate x² by subtracting 225 from both sides of the equation: x² = 1024 - 225, which simplifies to x² = 799. To find x, we take the square root of both sides of the equation: x = √799. Using a calculator, we find that the square root of 799 is approximately 28.26659. However, the problem asks for the answer to be given to one decimal place. Therefore, we round 28.26659 to one decimal place, which gives us 28.3. Thus, the horizontal distance between the wall and the bottom of the plank is approximately 28.3 inches. This step-by-step process demonstrates how to systematically solve the problem using the Pythagorean Theorem. It involves algebraic manipulation, calculation, and rounding to the required precision. By breaking down the solution into manageable steps, we can minimize errors and ensure accuracy. This approach not only helps in solving this specific problem but also provides a general framework for tackling similar mathematical challenges. The ability to follow a logical sequence of steps is crucial for problem-solving in mathematics and other disciplines. This exercise reinforces the importance of attention to detail and the accurate application of mathematical principles.

Final Answer and Conclusion

In conclusion, after applying the Pythagorean Theorem and following a step-by-step solution, we have determined that the horizontal distance between the wall and the bottom of the plank is approximately 28.3 inches, rounded to one decimal place. This problem demonstrates the practical application of the Pythagorean Theorem in a real-world scenario. By correctly identifying the sides of the right-angled triangle and setting up the appropriate equation, we were able to solve for the unknown distance. This exercise not only reinforces our understanding of the theorem but also highlights the importance of algebraic manipulation and careful calculation. The ability to visualize the problem, translate it into a mathematical model, and solve for the unknown is a valuable skill in mathematics and other fields. Understanding the Pythagorean Theorem and its applications is crucial for anyone studying geometry and related subjects. This problem provides a clear example of how mathematical concepts can be used to solve practical problems, enhancing our problem-solving abilities and critical thinking skills. The step-by-step approach we followed ensures accuracy and clarity in the solution process. By mastering these skills, we can confidently tackle similar challenges and apply mathematical principles to solve real-world problems effectively. The final answer, 28.3 inches, represents the horizontal distance and completes our solution to the plank and wall problem. This exercise underscores the importance of precision and attention to detail in mathematical calculations.