Problem 125 - A Graphical Approach to College Algebra

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$P=\frac{E^{2} R}{(r+R)^{2}} \text { for } R$$

Step-by-Step Solution

Answer

\( R = \frac{-Pr^2}{PR + 2Pr - E^2} \) after solving the equation.

Step 1: Write down the equation

We are given the equation \( P=\frac{E^{2} R}{(r+R)^{2}} \) and need to solve for \( R \). First, let's write it down clearly for reference.

Step 2: Clear the fraction

To eliminate the fraction, multiply both sides of the equation by \((r+R)^2\). This results in \(P(r+R)^2 = E^2 R\).

Step 3: Expand the left side

Expand \((r+R)^2\) to get \((r+R)^2 = r^2 + 2rR + R^2\). Substitute back into the equation: \(P(r^2 + 2rR + R^2) = E^2 R\).

Step 4: Distribute \( P \)

Distribute \( P \) on the left side: \( Pr^2 + 2PrR + PR^2 = E^2 R\).

Step 5: Rearrange terms

Move all terms involving \( R \) to one side: \( PR^2 + 2PrR - E^2 R = -Pr^2\).

Step 6: Factor out \( R \)

Factor \( R \) from the left side: \( R(PR + 2Pr - E^2) = -Pr^2\).

Step 7: Solve for \( R \)

Divide both sides by \((PR + 2Pr - E^2)\). This gives \( R = \frac{-Pr^2}{PR + 2Pr - E^2}\).

Key Concepts

Solving Formulas

Solving formulas involves isolating a particular variable on one side of the equation while keeping the equation balanced. It can seem like disentangling a complicated web to find one specific thread. In our case, we're asked to solve for the variable \( R \) in the formula \( P=\frac{E^{2} R}{(r+R)^{2}} \).The main strategy is to rearrange terms and eliminate any fractions or multiplicative factors to make \( R \) clear. Start by identifying the variable you need to isolate, and then use basic operations like addition, subtraction, multiplication, and division to manipulate the equation. When you encounter a fraction, as here, try to clear it early by multiplying both sides by the denominator. Each step brings you closer to isolating the variable, which will directly lead you to the solution you need.

Factoring

Factoring involves rewriting an expression as a product of its factors, making it easier to solve or simplify. This technique often appears when you're dealing with quadratic equations or expressions that can be regrouped into simpler multipliers.In the given exercise, after expanding terms and grouping them, we reached the step: \( R(PR + 2Pr - E^2) = -Pr^2 \). Here, factoring \( R \) from the left side helps in simplifying the equation.

• Factoring is useful because it reveals common factors that can be cancelled or divided out.
• It can also be a step towards using the zero-product property, which is handy in finding solutions to equations.

This manipulation clears the path for simplifying and finding the exact value or expression for \( R \), which ultimately leads to the solution of the formula.

Rearranging Equations

Rearranging equations means changing the order or structure of an equation to isolate a variable or make the equation easier to solve. This can involve moving terms from one side of the equation to the other and changing their signs.In the formula \( P(r^2 + 2rR + R^2) = E^2 R \), we start by distributing \( P \) and then gathering terms involving \( R \) on one side of the equation: \( PR^2 + 2PrR - E^2 R = -Pr^2 \).

• Rearranging helps in clearly laying out all terms with respect to the variable in question.
• Once the variable is isolated, you can then solve for it using arithmetic operations.

In our context, this method allows the equation to be refocused on \( R \), leading towards a straightforward division, which helps in finding \( R \). The step-by-step adjustment of equations is crucial in any problem that requires finding an unknown quantity.