Problem 46 - Algebra 1

Question

The formula \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\) relates the area and the dimensions of a trapezoid. Which way can you rewrite this relationship? (A) \(A=\frac{1}{2} h b_{1}+b_{2}\) (B) \(x=\frac{-y+5}{10}\) (C) \(h=\frac{A}{b_{1}+b_{2}}\) (D) \(b_{1}=2 A-b_{2}\)

Step-by-Step Solution

Answer

Option C, \(h=\frac{A}{b_{1}+b_{2}}\), is the correct way to rewrite the given formula \(A=\frac{1}{2} h(b_{1}+b_{2})\).

Step 1: Inspect the given options

Check each proposed equation to see if it correctly expresses the relationship between \(A\), \(h\), \(b_{1}\), and \(b_{2}\) established in the original formula.

Step 2: Evaluate option A

Looking at option A, \(A=\frac{1}{2} h b_{1}+b_{2}\), one can observe that it misplaces the parentheses, resulting in a different formula that doesn't correctly represent the relationship as stated in the question. Thus, option A is incorrect.

Step 3: Evaluate option B

Obviously option B, \(x=\frac{-y+5}{10}\), is completely unrelated to the original formula for the area of a trapezoid, as it even introduces new variables. Hence, option B is incorrect.

Step 4: Evaluate option C

Option C, \(h=\frac{A}{b_{1}+b_{2}}\), on the other hand, seems to result from isolating the variable \(h\) in the original formula. Dividing both sides of the equation by \(\frac{1}{2}(b_{1}+b_{2})\) gives \(h=\frac{A}{\frac{1}{2}(b_{1}+b_{2})}\), which simplifies to \(h=\frac{A}{b_{1}+b_{2}}\). Therefore, option C successfully rearranges the original formula, conserving the relationship between the variables.

Step 5: Evaluate option D

Evaluating option D, \(b_{1}=2 A-b_{2}\), also shows that it takes the original formula out of context, as the calculation doesn't match the original formula. Hence, option D isn't correct.

Key Concepts

Equation Manipulation

Equation manipulation involves changing the form of an equation to reveal different aspects of the relationship it represents. It's like rearranging the furniture in a room: you can see the same elements from a new perspective, helping you understand both the space and the components in it better.
You start with a formula, and through legal mathematical operations, such as adding, subtracting, multiplying, dividing, or factoring, you manipulate the equation to solve for different variables. For instance, given the trapezoid area formula \(A=\frac{1}{2} h(b_{1}+b_{2})\), equation manipulation allows us to express the formula differently by highlighting various variables.
Common goals of equation manipulation include:

• Rearranging formulas to solve for a specific variable.
• Simplifying complex expressions.
• Revealing hidden relationships among variables.

This skill becomes especially useful in real-world problems where you need to work with equations beyond their original scope. It provides flexibility to approach both academic exercises and practical challenges by addressing them from new angles.

Geometry

Geometry is the branch of mathematics that studies the sizes, shapes, properties, and dimensions of objects. It is fundamental in understanding figures like circles, triangles, and trapezoids, which are common in both theoretical math and practical applications like architecture and engineering.
The study of trapezoids involves understanding their unique properties:

• A trapezoid is a quadrilateral with at least one pair of parallel sides.
• The parallel sides are known as the bases, while the non-parallel sides are called the legs.
• The height of a trapezoid is the perpendicular distance between the bases.

For trapezoids, the area calculation incorporates both the height and the average length of the bases, as described by the formula \(A=\frac{1}{2} h(b_{1}+b_{2})\). This formula not only describes how the area is computed, but also emphasizes the relationship between the bases and the height. Understanding the formula is key to solving geometric problems.

Variable Isolation

Variable isolation is a technique used to solve equations for a particular unknown. It means rearranging an equation so that one variable stands alone on one side of the equation. This process is a crucial part of solving equations, allowing us to determine the value of one variable in terms of others.
When given a formula like the trapezoid area formula \(A=\frac{1}{2} h(b_{1}+b_{2})\), isolating a variable can help you solve for height \(h\), one of the bases, or any other part of the equation depending on what information you have and what you need to find. For example:

• If we need to find \(h\), we rearrange the formula to \(h=\frac{A}{b_{1}+b_{2}}\). Here, the variable \(h\) is isolated, showing how it's calculated.

This technique is particularly helpful in algebra and geometry, where relationships between variables are common. Each step in variable isolation must be mathematically sound to preserve the original relationship expressed by the equation.