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Chapter 9: Problem 40

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$A=\frac{1}{2} h\left(b_{1}+b_{2}\right) \quad \text { for } b_{2}$$

Short Answer

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

Step by step solution

01

- Write the formula

Start with the formula given in the problem: \[ A = \frac{1}{2} h (b_1 + b_2) \]
02

- Clear the fraction

Multiply both sides of the equation by 2 to eliminate the fraction: \[ 2A = h (b_1 + b_2) \]
03

- Divide by h

Divide both sides of the equation by h to isolate the term involving \( b_2 \): \[ \frac{2A}{h} = b_1 + b_2 \]
04

- Solve for \( b_2 \)

Subtract \( b_1 \) from both sides of the equation to solve for \( b_2 \): \[ b_2 = \frac{2A}{h} - b_1 \]

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Key Concepts

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

algebraic manipulation
Algebraic manipulation is a fundamental skill in mathematics that helps to solve equations by altering their structure. It involves using various techniques to simplify and rearrange equations to isolate a desired variable. In this exercise, we exemplify algebraic manipulation by solving for the variable \( b_2 \) in the formula \( A = \frac{1}{2} h (b_1 + b_2) \).
formulas in mathematics
Formulas in mathematics are equations that show the relationship between different variables. They are used to describe real-world phenomena and solve problems in various fields. In this exercise, we use the formula for the area of a trapezoid \( A = \frac{1}{2} h (b_1 + b_2) \). Formulas often require manipulation to solve for a specific variable, which we demonstrate here by isolating \( b_2 \).
solving for a variable
Solving for a variable involves manipulating an equation to get the desired variable by itself. This process often requires a combination of algebraic techniques like clearing fractions, distributing, and isolating terms. Let's look at the example provided: First, multiply both sides of the formula by 2 to get rid of the fraction: \[ 2A = h (b_1 + b_2) \]. Then, divide both sides by \( h \) to isolate \( b_1 + b_2 \): \[ \frac{2A}{h} = b_1 + b_2 \]. Finally, subtract \( b_1 \) from both sides: \[ b_2 = \frac{2A}{h} - b_1 \]. Following these steps methodically will help you solve for any variable in a formula.

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Most popular questions from this chapter

Perform the indicated operations and simplify. $$5 \cdot \frac{x}{10}$$

In Exercises \(52-55,\) simplify the given expression as completely as possible. Final answers should be expressed with positive exponents only. $$\frac{\left(x^{-2}\right)^{-5}}{x^{-2} x^{-5}}$$

Make up a verbal problem involving a ratio that would give rise to the following equation (be sure to indicate what the variable would represent): $$ \frac{x}{600}=\frac{9}{10} $$

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$y=m x+b \quad \text { for } x$$

In Exercises \(52-55,\) simplify the given expression as completely as possible. Final answers should be expressed with positive exponents only. $$x^{3} x^{4}\left(x^{3}\right)^{4}$$
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