Solving For Height Substitute Values Into Trapezoid Area Formula

Jul 7, 2025 by Admin 65 views

Substitute Values and Solve for the Unknown Variable

Understanding the Formula

In this mathematical exploration, we delve into the process of substituting given values into a formula and subsequently solving for the unknown variable. This is a fundamental skill in algebra and mathematics, applicable across various fields, including physics, engineering, and economics. The formula we will be working with is:

$A = \frac{1}{2} h(B + b)$

This formula represents the area of a trapezoid, where:

A is the area of the trapezoid
h is the height of the trapezoid
B is the length of one of the parallel sides (the base)
b is the length of the other parallel side (the other base)

Our task is to find the value of h, the height of the trapezoid, given the values of A, B, and b. Understanding the formula and the variables it involves is the first crucial step in successfully solving the problem. Before we dive into the substitution and solving process, let's break down the formula further. The formula essentially states that the area of a trapezoid is equal to one-half multiplied by the height, and then multiplied by the sum of the lengths of the two parallel sides. This can be visualized as finding the average length of the two parallel sides ((B+b)/2) and then multiplying it by the height (h). This provides a clear geometrical interpretation of the formula. Now that we have a solid understanding of the formula, we can move on to the next stage, which involves substituting the given values into the appropriate places in the formula. This step is crucial as it sets the foundation for the subsequent algebraic manipulations that will lead us to the solution for the unknown variable, h. Remember, accuracy in substitution is paramount; a small error at this stage can propagate through the entire solution process, leading to an incorrect answer. Therefore, we will proceed with meticulous care, ensuring that each value is placed correctly before moving on to the next step.

Substituting the Given Values

Now, let's substitute the given values into the formula. We are given:

A = 26
B = 8
b = 5

Plugging these values into the formula, we get:

$26 = \frac{1}{2} h(8 + 5)$

This equation now contains only one unknown, h, which we can solve for. Substitution is a critical step in solving algebraic equations and mathematical problems. It involves replacing variables with their corresponding numerical values, allowing us to transform a general formula into a specific equation that can be solved for the remaining unknown. In this case, by substituting the values for A, B, and b, we have effectively transformed the area formula for a trapezoid into an equation that specifically represents the trapezoid in question. This transformation is crucial because it allows us to isolate and determine the value of the unknown variable, which in this instance is the height, h. Before proceeding further, let's take a moment to review the substitution process to ensure accuracy. Double-checking each substitution helps to minimize the risk of errors that could lead to an incorrect solution. Once we are confident that the substitutions are correct, we can proceed to simplify the equation and begin the process of solving for h. The next step will involve simplifying the expression within the parentheses and then addressing the fraction, all in the pursuit of isolating h on one side of the equation. Remember, the goal is to manipulate the equation in a way that preserves equality while bringing us closer to determining the numerical value of the height of the trapezoid.

Solving for h

To solve for h, we need to isolate it on one side of the equation. Let's follow these steps:

Simplify the expression inside the parentheses: $26 = \frac{1}{2} h(13)$
Multiply both sides of the equation by 2 to get rid of the fraction: $2 * 26 = h(13)$ $52 = 13h$
Divide both sides by 13 to isolate h: $\frac{52}{13} = h$ $h = 4$

Therefore, the value of h is 4. Solving for an unknown variable in an equation is a fundamental skill in algebra and mathematics. It involves applying a series of algebraic manipulations to isolate the variable on one side of the equation, thereby determining its value. In this case, we followed a systematic approach to solve for h, the height of the trapezoid. First, we simplified the expression within the parentheses to make the equation more manageable. Next, we addressed the fraction by multiplying both sides of the equation by 2. This step is crucial because it eliminates the fraction, making the equation easier to work with. After eliminating the fraction, we were left with a simple equation, 52 = 13h. To isolate h, we divided both sides of the equation by 13. This operation maintains the equality of the equation while effectively isolating h on one side. Finally, we arrived at the solution, h = 4. This value represents the height of the trapezoid, given the area and the lengths of the parallel sides. It is important to note that each step in the solving process is based on fundamental algebraic principles. These principles ensure that the equality of the equation is maintained throughout the manipulation process. By following these principles, we can confidently arrive at the correct solution for the unknown variable.

Final Answer

Therefore, by substituting the given values into the formula and solving for the unknown variable, we find that:

$h = 4$

This comprehensive process demonstrates the power of algebraic manipulation in solving real-world problems. This result provides the height of the trapezoid, completing our solution. In summary, we started with the formula for the area of a trapezoid, substituted the provided values for the area and the lengths of the parallel sides, and then systematically solved for the height. This process not only demonstrates a specific solution but also underscores the importance of understanding algebraic principles in solving mathematical problems. The ability to substitute values into formulas and solve for unknowns is a fundamental skill that is applicable across various disciplines, from physics and engineering to economics and finance. By mastering this skill, students and professionals alike can effectively tackle a wide range of problems that involve mathematical relationships. Moreover, the systematic approach we employed in this solution highlights the importance of careful and methodical problem-solving. Each step was carefully executed, ensuring that we maintained the equality of the equation and ultimately arrived at the correct solution. This approach can be applied to various mathematical problems, fostering accuracy and confidence in problem-solving abilities. Finally, this exercise serves as a reminder of the interconnectedness of mathematical concepts. The area of a trapezoid, algebraic manipulation, and problem-solving strategies all come together in this example to illustrate the beauty and utility of mathematics in understanding and solving real-world problems. With a solid understanding of these concepts, one can approach mathematical challenges with clarity and competence, making informed decisions and achieving accurate results.