Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(h: A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\)

Step-by-Step Solution

Answer

The height \( h \) is given by \( h = \frac{2A}{b_1 + b_2} \).

Step 1: Understand the formula and identify the variable to solve for

We are given the formula for the area of a trapezoid, \( A = \frac{1}{2} h (b_1 + b_2) \), where \( A \) is the area, \( h \) is the height, and \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides. Our task is to solve for \( h \).

Step 2: Isolate the term containing the variable

Start by isolating the term that contains \( h \). Multiply both sides of the equation by 2 to eliminate the fraction: \( 2A = h(b_1 + b_2) \).

Step 3: Solve for the variable

Next, solve for \( h \) by dividing both sides by \((b_1 + b_2)\): \( h = \frac{2A}{b_1 + b_2} \). This gives us the formula to find the height \( h \).

Key Concepts

Understanding the Trapezoid Area Formula

The trapezoid area formula is essential for calculating the area of a trapezoid, which is a quadrilateral with exactly one pair of parallel sides. The formula is as follows: \[ A = \frac{1}{2} h (b_1 + b_2) \] In this formula:

• \( A \) represents the area of the trapezoid.
• \( h \) is the height, which is the perpendicular distance between the two parallel sides.
• \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides.

This formula works by essentially calculating the average length of the parallel sides and multiplying by the height. It is helpful to visualize the trapezoid and understand its parts to see how the area coverage is calculated. This foundational equation allows solving for any unknown value when plugged accordingly.

Using Algebra to Rearrange Equations

Algebra is a powerful mathematical tool that allows us to manipulate equations and solve for unknown variables. In many cases, the goal is to rearrange the given equation to make one specific variable the subject. In the given trapezoid area example, our task was to solve for \( h \) in the formula:\[ A = \frac{1}{2} h (b_1 + b_2) \] To do this using algebraic techniques, we took these simple manipulative steps:

• Multiply both sides by 2, to remove the fraction: \[ 2A = h (b_1 + b_2) \]
• Next, divide both sides by the entire expression \((b_1 + b_2)\) to isolate \( h \): \[ h = \frac{2A}{b_1 + b_2} \]

These steps are typical in algebra, helping us to rearrange and understand various formulas. Algebra provides the flexibility needed to solve equations, making it an essential skill in mathematics.

Isolating the Variable: Solving for Height

Isolating a variable involves rearranging an equation to get the variable of interest by itself on one side of the equation. This process is crucial when dealing with problems where the equation is given but one specific part needs to be solved for, as seen in our trapezoid problem. To find the height \( h \), we took a careful approach:

• Firstly, we had the equation \( A = \frac{1}{2} h (b_1 + b_2) \).
• The primary step to isolation is to remove any coefficients or grouped terms from the variable. Multiplying by 2 helped us get rid of the fraction.
• Subsequently, we divided by the grouped term \((b_1 + b_2)\), effectively leaving \( h \) isolated: \[ h = \frac{2A}{b_1 + b_2} \]

This procedure is a common algebraic approach and can be applied to various equations, enabling us to solve for any variable as required. By consistently applying these steps, you can demystify more complex algebraic equations easily.