Problem 71 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(A=\frac{1}{2}\left(b_{1}+b_{2}\right) h\) for \(b_{1}\)

Step-by-Step Solution

Answer

\(b_1 = \frac{2A}{h} - b_2\).

Step 1: Understand the Formula

The given formula is the area of a trapezoid: \(A=\frac{1}{2}(b_{1}+b_{2})h\), where \(A\) is the area, \(b_1\) and \(b_2\) are the lengths of the two bases, and \(h\) is the height. We need to solve for \(b_1\).

Step 2: Multiply Both Sides by 2

To eliminate the fraction, multiply both sides of the equation by 2: \(2A = (b_1 + b_2)h\). This step helps us isolate the terms containing \(b_1\).

Step 3: Divide Both Sides by the Height

To further isolate the bases, divide both sides by \(h\): \(\frac{2A}{h} = b_1 + b_2\). This puts \(b_1\) and \(b_2\) alone on one side.

Step 4: Subtract \(b_2\) from Both Sides

To solve for \(b_1\), subtract \(b_2\) from both sides: \(\frac{2A}{h} - b_2 = b_1\). Now, \(b_1\) is isolated.

Key Concepts

Trapezoid Area Formula

A trapezoid is a four-sided figure with one pair of parallel sides. In geometry, calculating the area of a trapezoid involves its unique shape, and is done using the trapezoid area formula. The formula for the area of a trapezoid is expressed as:\[ A = \frac{1}{2} (b_1 + b_2)h \]where:

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

This formula is based on the average length of the bases, \((b_1 + b_2)/2\), multiplied by the height, which is a similar process to finding the area of a rectangle. Understanding this formula is crucial because it applies to various geometric problems involving trapezoids.

Isolating Variables

When we want to solve an equation for a particular variable, we need to isolate that variable on one side of the equation. In the context of our problem, we start with the trapezoid area formula:\[ A = \frac{1}{2} (b_1 + b_2)h \]Our goal is to solve for \(b_1\). Isolating variables involves algebraic manipulation to rearrange the equation until the desired variable stands alone. To achieve this, we deploy sequential steps:

• First, we eliminate any fractions by multiplying all terms to clear the denominator.
• Secondly, we move all terms not containing the desired variable to the other side by adding, subtracting, dividing, or multiplying as needed.

The concept of isolating variables is pivotal in algebra and allows us to solve for unknowns effectively.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to make them easier to work with or to solve for specific variables. In our exercise, we applied algebraic manipulation to deal with the trapezoid area formula, especially focusing on the variable \(b_1\).Here, we:

• Multiply both sides of the formula by 2 to eliminate the fraction and simplify the expression:
  \[ 2A = (b_1 + b_2)h \]
• Next, divide both sides by the height \(h\) to further isolate the sum of the bases:
  \[ \frac{2A}{h} = b_1 + b_2 \]
• Finally, we subtract \(b_2\) on one side to solve directly for \(b_1\) :
  \[ b_1 = \frac{2A}{h} - b_2 \]

These steps illustrate how algebraic manipulation helps transform equations and find explicit solutions for variables.

Geometry Applications

Understanding geometry and the application of formulas within that realm provides an essential foundation for solving practical problems. The trapezoid area formula and the algebraic processes used to solve for specific variables can be applied in numerous scenarios. Some examples include:

• Architects calculating surface areas for design structures.
• Mathematicians solving theoretical problems involving planes and slopes.
• Engineers determining materials needed for trapezoidal components.

Proficiency in these applications ensures we can apply mathematical solutions in real-world contexts, helping to visualize and implement effective designs and solutions.