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Chapter 2: Problem 67

Solve for the specified variable or expression. $$ A=\frac{1}{2} h(b+d) \text { for } b $$

Short Answer

Expert verified
The solution for \( b \) is \( b = \frac{2A}{h} - d \).

Step by step solution

01

Identify the Given Equation

The given equation is the formula for the area of a trapezoid: \( A = \frac{1}{2}h(b+d) \). We need to solve this equation for \( b \).
02

Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by 2: \( 2A = h(b+d) \).
03

Isolate the Variable Term

Divide both sides of the equation by \( h \) to isolate the term with \( b \): \( b + d = \frac{2A}{h} \).
04

Solve for the Variable

Subtract \( d \) from both sides to solve for \( b \): \( b = \frac{2A}{h} - d \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trapezoid Area Formula
The area of a trapezoid is calculated using a specific formula that takes into account the trapezoid's unique shape. A trapezoid has two parallel sides called the bases, and the formula for its area is formed based on these. The formula is given by: \[ A = \frac{1}{2} h(b + d) \] where:
  • \( A \) is the area,
  • \( h \) is the height of the trapezoid,
  • \( b \) and \( d \) are the lengths of the two parallel sides (the bases).
This formula effectively averages the lengths of the two bases and then multiplies by the height, and finally by a half, to give you the area of the trapezoid. This is because, unlike a rectangle, the parallel sides are not the same length, so we take their average to find a representative length for calculating the area.
Linear Equations
Linear equations are fundamental in algebra and mathematics. They involve variables that appear with a power of one, which means they graph as straight lines. In the area of solving for a variable, understanding linear equations is key. A linear equation can be represented in the standard form: \[ ax + by = c \] Where:
  • \( a, b, c \) are constants,
  • \( x, y \) are variables.
The goal often involves isolating one variable, making it dependent on the other(s) and any constants. This isolation helps in solving real-world problems by representing relationships between quantities.
Variable Isolation
Variable isolation is a crucial concept in solving equations and deriving formulas for specific variables. It means isolating the variable of interest on one side of the equation. In the given trapezoid area formula example, the task was to isolate the variable \( b \). Here鈥檚 how variable isolation works generally:
  • Start by simplifying both sides of the equation if necessary, like removing fractions through multiplication.
  • Move other terms involving the variable to one side by performing inverse operations such as addition/subtraction or multiplication/division.
  • Solve for the variable by isolating it completely on one side of the equation.
This process is handy when solving real-life problems where you know a formula but need to find a specific component, such as in physics or engineering disciplines. Understanding and practicing variable isolation can simplify complex equations and make solving them intuitive.

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