Problem 102 - College Algebra Essentials

Question

Solve each equation. $$ \text { Solve for } A: r=\sqrt{\frac{A}{4 \pi}} $$

Step-by-Step Solution

Answer

The value of \(A\) is \(4\pi r^2\).

Step 1: Square both sides

Since we have to isolate \(A\), and \(A\) is under a square root on the right side, we need to eliminate the square root. This can be done by squaring both sides. Doing so will give us\[r^2 = \frac{A}{4\pi}.\]

Step 2: Solve for \(A\)

Next, we need to isolate \(A\) on one side of the equation. This could be achieved by multiplying both sides of the equation by \(4\pi\). We have\[A = 4\pi r^2.\]

Key Concepts

Square Root Elimination

When dealing with equations involving square roots, the initial step often involves eliminating the square root to simplify the equation. In the exercise provided, the variable \( A \) is inside a square root expression \( \sqrt{\frac{A}{4 \pi}} \). To eliminate the square root, you need to square both sides of the equation. This means you perform the operation where you multiply each side by itself. Here’s how it looks with our example:

• Given: \( r = \sqrt{\frac{A}{4 \pi}} \)
• Square both sides: \( r^2 = \frac{A}{4 \pi} \)

Squaring both sides effectively removes the square root, as the square and square root cancel each other out. This transformation is crucial, as it leaves you with a simpler equation where \( A \) is no longer "trapped" under the square root.

Algebraic Manipulation

After the square root is eliminated and you have the equation \( r^2 = \frac{A}{4 \pi} \), the next step is algebraic manipulation. This process involves rearranging the equation to effectively solve for the desired variable by performing operations like multiplication, division, addition, or subtraction.

In our scenario, the goal is to clear \( A \) from its current fraction form. The equation has \( A \) divided by \( 4 \pi \). To get rid of this fraction, you need to multiply both sides by \( 4 \pi \). This is what happens:

• Multiply both sides by \( 4 \pi \): \( 4 \pi \cdot r^2 = A \)

Multiplying by \( 4 \pi \) cancels the division, thus moving \( A \) out of the fraction and leaving it alone on the right side. This manipulation brings you one step closer to isolating and solving for \( A \).

Isolation of Variable

The final step in solving equations like the one in the exercise is the isolation of the variable you want to solve for, in this case, \( A \). Thanks to the operations performed earlier, \( A \) is already isolated. You have manipulated the initial equation to end up with:

• \( A = 4 \pi r^2 \)

This expression shows \( A \) as the subject of the equation, meaning \( A \) is by itself on one side. This completes the solution process, offering a direct expression for \( A \) in terms of \( r \).

Making \( A \) the subject confirms that you have successfully isolated it, which allows you to calculate \( A \) directly if \( r \)'s value is known.