Problem 65 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} \(\mathscr{A}=\frac{1}{2} h\left(b_{1}+b_{2}\right)\) for \(h \quad\) (Area of a trapezoid)

Step-by-Step Solution

Answer

The formula for \(h\) is \(h = \frac{2 \mathscr{A}}{b_{1} + b_{2}}\).

Step 1: Understand the Formula

You are given the formula for the area of a trapezoid: \(\mathscr{A}=\frac{1}{2} h\left(b_{1}+b_{2}\right)\). This formula expresses the area (\(\mathscr{A}\)) in terms of the height (\(h\)) and the lengths of the two bases (\(b_1\) and \(b_2\)). You need to solve for the height (\(h\)).

Step 2: Multiply Both Sides by 2

To eliminate the fraction, multiply both sides of the equation by 2: \[2 \times \mathscr{A} = 2 \times \left( \frac{1}{2} h (b_{1}+b_{2}) \right)\]This simplifies to: \[2 \mathscr{A} = h (b_{1} + b_{2})\]

Step 3: Divide Both Sides by \((b_{1} + b_{2})\)

To solve for \(h\), divide both sides of the equation by \((b_{1} + b_{2})\): \[h = \frac{2 \mathscr{A}}{b_{1} + b_{2}}\]

Step 4: Final Formula for \(h\)

You now have the formula for \(h\). This allows you to find the height of the trapezoid when you know the area and the lengths of the two bases: \[h = \frac{2 \mathscr{A}}{b_{1} + b_{2}}\].

Key Concepts

Area of Trapezoid

A trapezoid is a four-sided figure with at least one pair of parallel sides. The area of a trapezoid is the space contained within its boundaries. To find the area, we use the formula: \[ \mathscr{A} = \frac{1}{2} h(b_{1} + b_{2}) \]Here:

• \(\mathscr{A}\) represents the area of the trapezoid.
• \(h\) is the height, which is the perpendicular distance between the parallel sides.
• \(b_{1}\) and \(b_{2}\) are the lengths of the two parallel sides, also known as the bases.

To compute the area, multiply the sum of the bases \((b_{1} + b_{2})\) by the height \(h\), then divide the product by 2. This formula is an easy and efficient way to calculate the area if the dimensions are known.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions to solve for variables or to simplify expressions. In the problem with the trapezoid, we have to manipulate the given formula to solve for the height \(h\). This process involves several steps:- **Eliminate Fractions:** Multiplying both sides of the equation by 2 cancels out the fraction \(\frac{1}{2}\). This simplifies the equation, making it easier to deal with.- **Rearrange Terms:** Once the fraction is removed, the equation becomes \(2\mathscr{A} = h(b_{1} + b_{2})\). This positions the terms so that we can easily isolate \(h\).- **Isolate the Desired Variable:** To solve for \(h\), we divide both sides by \((b_{1} + b_{2})\). This step is crucial to get \(h\) on one side of the equation.These steps of manipulation help us derive the clear formula for \(h\), which can then be used for calculations.

Solving for a Variable

Solving for a variable means isolating that variable on one side of the equation, allowing us to express it in terms of other known quantities. In the formula for the trapezoid's area, our goal was to solve for the height \(h\):- **Understand the Goal:** Our target is to have \(h\) by itself on one side of the equation.- **Employ Algebraic Techniques:** Use simple algebraic actions such as multiplication or division to manipulate the equation.- **Express the Variable Clearly:** After division in the given problem, we find \(h = \frac{2\mathscr{A}}{b_{1} + b_{2}}\).This process of variable isolation is fundamental in algebra. It is widely used to rearrange formulas in practical problem-solving scenarios, enabling us to substitute known values in to discover unknowns.