Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(h: A=\frac{1}{2} b h\)

Step-by-Step Solution

Answer

The height \(h\) is \(h = \frac{2A}{b}\).

Step 1: Understand the Formula

The given formula is for the area of a triangle, where \(A\) is the area, \(b\) is the base, and \(h\) is the height. The formula is \(A = \frac{1}{2} b h\). We need to solve this equation for \(h\).

Step 2: Isolate the Variable (h)

To solve for \(h\), we need to get \(h\) by itself on one side of the equation. Start by cancelling out the fraction \(\frac{1}{2}\). Multiply both sides of the equation by 2 to get \(2A = bh\).

Step 3: Solve for h

Next, divide both sides of the equation by \(b\) to isolate \(h\). This gives us \(h = \frac{2A}{b}\). The height \(h\) is now expressed in terms of \(A\) and \(b\).

Key Concepts

Area of a Triangle

The area of a triangle is a measure of the region enclosed by the three sides of a triangle. Understanding this basic concept is crucial because it applies to various geometrical problems and real-world tasks. The formula for the area of a triangle is expressed as \( A = \frac{1}{2} b h \), where:

• \(A\) is the area of the triangle
• \(b\) is the base
• \(h\) is the height

This formula states that the area is half the product of the base and height. The base \( b \) can be any one of the triangle's sides, but it's usually the side that lies horizontally.
The height \( h \), or altitude, is the perpendicular distance from the chosen base to the opposite vertex.

Calculating the area by using this formula requires knowing both the base and the height. It’s important to ensure these measurements are perpendicular to each other to use the formula correctly.

Isolating the Variable

Isolating the variable is a fundamental step in solving equations. In our exercise, we want to solve for \( h \) in the triangle's area formula. Isolating a variable means getting the variable alone on one side of the equation. Here’s how we do it:

• First, observe the equation \( A = \frac{1}{2} b h \).
• To remove the fraction, multiply both sides of the equation by 2. This gives \( 2A = bh \).

This step effectively 'undoes' the multiplication by \( \frac{1}{2} \), making it simpler to see how to get \( h \) by itself.
Once the fraction is cleared, the equation becomes easier to manipulate, setting the stage for isolating \( h \). By performing these operations correctly, you ensure that \( h \) is the focal point, which is crucial to subsequent problem-solving steps.

Algebraic Manipulation

Algebraic manipulation is a process used to rearrange equations in order to solve for a specific variable. In our context, it involves transforming the equation \( 2A = bh \) to solve for \( h \). Here's how:

• Recognize that \( h \) is multiplied by \( b \) in the equation \( 2A = bh \).
• To isolate \( h \), divide both sides by \( b \). This yields the equation \( h = \frac{2A}{b} \).

This final step of dividing ensures that \( h \) is alone on one side, expressed in terms of \( A \) and \( b \).
Algebraically, it's vital to perform the same operation on both sides of the equation to maintain equality.
Successful algebraic manipulation allows you to rewrite formulas to solve for different variables, adapting standard formulas to specific needs. This flexibility is one of the strengths of algebra in problem-solving.