Problem 39 - College Algebra

Question

Solve for \(h : A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula

The given formula is an equation that expresses the area \(A\) of a trapezoid. Here, \(h\) represents the height, and \(b_1\) and \(b_2\) are the lengths of the two parallel sides (bases). We need to solve this equation for \(h\).

Step 2: Eliminate the Fraction

The equation is \( A = \frac{1}{2} h(b_1 + b_2) \). To eliminate the fraction, multiply both sides of the equation by 2 to get rid of the \(\frac{1}{2}\). Thus, we have: \[ 2A = h(b_1 + b_2) \]

Step 3: Solve for \(h\)

Now that the equation is \(2A = h(b_1 + b_2)\), we want to isolate \(h\). To do this, divide both sides of the equation by \((b_1 + b_2)\):\[ h = \frac{2A}{b_1 + b_2} \] This formula gives us the height \(h\) in terms of the area \(A\) and the lengths of the bases \(b_1\) and \(b_2\).

Key Concepts

Geometry

Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. Understanding the geometric shapes is crucial, especially when working with formulas involving these figures. A trapezoid is a flat, four-sided shape with at least one pair of parallel sides, known as the bases (\(b_1\) and \(b_2\)). The distance between the bases is referred to as the height (\(h\)), which is perpendicular to the bases. Trapezoids are unique because they can vary in shape, having no equal sides or angles other than their parallel sides being equal in measure to each base. This geometric property directly relates to the formula for the area, as it combines these base lengths and the height to calculate the area effectively. Understanding this link between the shape and its area formula provides a deeper insight into the spatial configuration of trapezoids.

Solving Equations

Solving equations involves finding the value of an unknown that makes an equation true. When dealing with the formula for the area of a trapezoid, it is necessary to solve for \(h\), the height, given that all other values in the equation are known. By interpreting the given equation, \(A = \frac{1}{2} h (b_1 + b_2)\), one recognizes that it is a basic linear equation. The process of solving equations generally requires logical steps to isolate the unknown variable. Key steps include:

• Understanding the equation components and their relationships.
• Eliminating fractions by multiplying through to clear any denominators.
• Rearranging the equation to isolate the variable of interest.

In this exercise, multiplying both sides by 2 dissolves the fraction, simplifying the process of solving the equation for the desired variable, \(h\).

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to isolate a specific variable. It involves using mathematical operations to change the form of the equation without altering the equality. For the trapezoid area formula, we needed to solve for \(h\), which required careful manipulation of the initial equation \(A = \frac{1}{2} h (b_1 + b_2)\). Here is a step-by-step explanation of the manipulation needed:

• Initially, the equation involves a fraction, so we multiply both sides by 2 to clear the fraction, obtaining \(2A = h (b_1 + b_2)\).
• Next, we aim to isolate \(h\). We divide both sides of the equation by \((b_1 + b_2)\), leading us to the expression \(h = \frac{2A}{b_1 + b_2}\).
• This results in the height being expressed as a function of area and base lengths, which is crucial for solving practical geometry problems involving trapezoids.

Algebraic manipulation is a vital skill that enables mathematicians and students alike to navigate through equations efficiently, ensuring well-informed solutions.