Problem 70 - Intermediate Algebra

Question

Solve each equation for the specified variable or expression. See Example 9. $$ R_{1}=\sqrt{\frac{A}{\pi}-R_{2}^{2}} \text { for } A $$

Step-by-Step Solution

Answer

The expression for \( A \) is \( A = \pi(R_1^2 + R_2^2) \).

Step 1: Square Both Sides

The given equation is \( R_1 = \sqrt{\frac{A}{\pi} - R_2^2} \). Start by eliminating the square root on the right by squaring both sides of the equation. This gives: \( R_1^2 = \frac{A}{\pi} - R_2^2 \).

Step 2: Isolate the Fraction

Next, solve for \( \frac{A}{\pi} \) by adding \( R_2^2 \) to both sides: \( R_1^2 + R_2^2 = \frac{A}{\pi} \).

Step 3: Solve for A

To isolate \( A \), multiply both sides by \( \pi \), resulting in: \( A = \pi(R_1^2 + R_2^2) \).

Key Concepts

Solving equations

Solving equations is like solving a puzzle. We are working to find the value, or values, of a variable that makes the equation true. Start by examining what we currently know and what we need to find. In this case, we need to solve for the variable \( A \) in the given equation. The key steps involve performing operations that keep the equation balanced—whatever you do to one side, you must do to the other. This might include adding, subtracting, multiplying, or dividing both sides by the same number or expression to gradually get closer to the solution.

• Identify the variable you need to solve for.
• Perform operations to isolate and solve for the variable.
• Keep equations balanced by doing the same thing to both sides.

Each operation brings us closer to the solution, unraveling the puzzle one piece at a time.

Isolating variables

Isolating a variable means getting it by itself on one side of the equation. This often involves inverse operations, or doing the opposite of what's being done to the variable, to simplify the equation. In our example, we started by eliminating the square root by squaring both sides to ease the isolation of \( A \). After squaring, we focus on moving terms around to have terms with \( A \) on one side, leaving it isolated. Here’s a basic strategy for isolating variables:

• Identify the operations acting on the variable.
• Use inverse operations to cancel them out—addition undoes subtraction, multiplication undoes division, etc.
• Simplify the equation step-by-step, keeping the variable as the focus.

By systematically isolating the variable, a once complex equation becomes much simpler.

Square roots

Square roots are mathematical operations that essentially ask what number, when multiplied by itself, equals the given number. In equations, square roots often appear when dealing with quadratic relationships. In our example, to solve for \( A \), the first step was squaring both sides of the equation to eliminate the root. This transformation helps in simplifying the equation significantly. Remember these tips when working with square roots:

• Squaring a square root simply cancels it out: \( \sqrt{x} \cdot \sqrt{x} = x \).
• Be cautious with negative numbers—squaring a negative yields a positive, which can influence the solution.
• When both sides of an equation involve a square root, squaring can help simplify and solve it.

These steps unlock the solution by dismantling the complexities introduced by square roots.