Problem 84 - Beginning and Intermediate Algebra

Question

Solve. \(s=\sqrt{\frac{A}{6}},\) for \(A\) (A geometry formula)

Step-by-Step Solution

Answer

The short version of the answer is: \(A = 6s^2\)

Step 1: Identify the formula and the problem

We must isolate A in the given equation: \(s=\sqrt{\frac{A}{6}}\)

Step 2: Square both sides

To eliminate the square root, we will square both sides of the equation: \[(s)^{2}=\left(\sqrt{\frac{A}{6}}\right)^{2}\]

Step 3: Simplify the equation

Squaring both sides of the equation, we get: \[s^{2} = \frac{A}{6}\]

Step 4: Solve for A

Now, to solve for A, multiply both sides of the equation by 6: \[A = 6s^{2}\] Now, we have found the value of A in terms of s:

Step 5: Final Answer

\( A = 6s^{2} \)

Key Concepts

Geometry Formula

Geometry often provides us with formulas that relate different dimensions or measurements. In the problem at hand, we started with a geometric formula:

• \( s = \sqrt{\frac{A}{6}} \)

This equation connects three elements:

• The variable \(s\), which might represent a side length or another linear measurement.
• The variable \(A\), which could denote an area or a similar dimensional property.
• The constant 6, which is a factor in the relationship between \(s\) and \(A\).

Such formulas are crucial in geometry as they allow us to relate and understand different parts of geometric figures. The goal is to comprehend how these variables influence each other through their measurements.

Algebraic Manipulation

Algebraic manipulation involves reorganizing and simplifying equations to solve for a specific variable. In our equation \(s = \sqrt{\frac{A}{6}}\), the initial goal was to solve for \(A\).Here is a step-by-step rundown:

• First, identify the operation that connects the variables and recognize if any additional operations (such as square roots or division) must be undone.
• Next, use inverse operations to counteract these. For example, squaring is the inverse of taking a square root, which comes in handy to simplify the problem.
• Once the equation has been simplified by removing complicated elements, the next step typically involves rearranging the terms to isolate the desired variable. In this case, the formula was manipulated from \(s = \sqrt{\frac{A}{6}}\) to eventually arrive at \(A = 6s^2\).

Algebraic manipulation simplifies the relationships between variables, allowing us to derive new equations and solve for unknowns effectively.

Square Root Elimination

A critical step in solving the equation was the elimination of the square root, a technique often used in algebra to simplify expressions. To effectively eliminate the square root from \(s = \sqrt{\frac{A}{6}}\), we followed these steps:

• Square both sides of the equation to remove the square root. This yields \(s^2 = \left(\sqrt{\frac{A}{6}}\right)^2\), simplifying to \(s^2 = \frac{A}{6}\).
• Once the square root is eliminated, the equation becomes easier to work with, allowing further algebraic manipulation.
• Always remember that squaring both sides of an equation can introduce extra solutions (extraneous solutions), so it's essential to verify your results in the original context of the problem.

By removing square roots, you transform challenging equations into more straightforward algebraic forms, making it easier to solve for unknowns such as \(A\). This step is essential in many algebraic and geometric problems.