Problem 40 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ A=\frac{1}{2} h(a+b), \text { for } b $$

Step-by-Step Solution

Answer

\( b = \frac{2A}{h} - a \)

Step 1: Identify the Given Formula

We are given the formula for the area of a trapezoid: \[ A = \frac{1}{2} h(a+b) \]and need to solve for the variable \( b \).

Step 2: Isolate the Term Containing Variable

To solve for \( b \), we first isolate the term \( (a+b) \) by eliminating the fraction. Multiply both sides by 2 to get:\[ 2A = h(a+b) \].

Step 3: Solve for \( a+b \)

Divide both sides of the equation by \( h \) to solve for \( (a+b) \):\[ \frac{2A}{h} = a+b \].

Step 4: Isolate the Variable \( b \)

Now, subtract \( a \) from both sides to solve for \( b \):\[ b = \frac{2A}{h} - a \].

Step 5: Final Expression for \( b \)

We have expressed \( b \) in terms of \( A, h, \) and \( a \):\[ b = \frac{2A}{h} - a \].

Key Concepts

Area of a Trapezoid

The area of a trapezoid is an important concept in geometry. A trapezoid is a four-sided figure with at least one pair of parallel sides. These parallel sides are called bases. The formula to calculate the area of a trapezoid is:

• \( A = \frac{1}{2} h(a+b) \)

In this formula:

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

To find the area, you add the lengths of the two bases, multiply by the height, and then divide by two. This formula emphasizes that the area of a trapezoid is a combination of the areas under each base.It is a straightforward adaptation from calculating the area of rectangles or parallelograms but accounting for the different lengths of the bases.

Isolating Variables

Isolating a variable in an equation is a fundamental algebraic technique. It involves rearranging the equation to express a specific variable independently on one side. This skill is essential for solving equations and understanding how changes in one element affect another.When we isolate a variable, we essentially perform operations to strip away all other terms and coefficients. Let's break it down:

• Start with the equation in its original form.
• Identify terms involving other variables or constants that accompany the variable of interest.
• Use algebraic operations such as addition, subtraction, multiplication, or division to move these accompanying elements to the opposite side of the equation.

In our trapezoid area example, isolating \( b \) required multiplying both sides by 2, then dividing by \( h \), and finally subtracting \( a \). Ensuring that \( b \) was left alone demonstrates how the value of \( b \) depends solely on the remaining variables.

Algebraic Manipulation

Algebraic manipulation involves performing valid operations on equations to simplify them or solve for a particular variable. It's like transforming a messy problem into something clearer.Several rules guide algebraic manipulation:

• Whatever operation you perform on one side of the equation, you must also perform on the other side to keep the equation balanced.
• Common operations include distributing, factoring, expanding, and combining like terms.
• Carefully follow all algebraic rules to avoid losing alignment with the equation's initial conditions.

For example, in solving for \( b \), we manipulated the equation \( A = \frac{1}{2} h(a+b) \) by eliminating the fraction. This step involved multiplying both sides by 2 and dividing by \( h \) to isolate \( a+b \). By systematically applying operations, we transitioned to the simpler form, \( b = \frac{2A}{h} - a \). Each step of algebraic manipulation is vital to maintain integrity as you focus on simplifying the expression for your target variable.