Problem 78 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for the indicated variable. Assume no denominators are \(0 .\) $$S=2 \pi r h+2 \pi r^{2}, \quad \text { for } r$$

Step-by-Step Solution

Answer

The variable \( r \) is given by the formula \( r = \frac{ S}{2 \pi (h + r ) } \).

Step 1: Factor out the common term

The given equation is: \[S=2 \pi r h+2 \pi r^{2} \] Factor out the common term of \(2 \pi r \) from the right-hand side of the equation: \[S=2 \pi r(h + r)\]

Step 2: Isolate the variable \( r \)

To solve for \( r \), divide both sides of the equation by \(2 \pi (h + r)\): \[ \frac{S}{{2 \pi ( h + r) }} = r \] This puts \( r \) on one side of the equation.

Step 3: Recap the isolated variable

The isolated variable \( r \) in terms of the other constants is: \[ r = \frac{ S }{ 2 \pi ( h + r ) } \] Therefore, the isolated formula for \( r \) is obtained.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the foundation of solving equations. It involves rearranging and simplifying the equation to make it easier to work with. In this exercise, we start with the equation: \[ S = 2 \pi r h + 2 \pi r^{2} \]

• Identify common terms or factors.
• Apply basic operations such as addition, subtraction, multiplication, and division.

In our example, we can see that both terms on the right-hand side share a common factor, \(2 \pi r \). Recognizing this is crucial because it allows us to factor that term out, simplifying the equation. By factoring, we consolidate similar terms, reducing complexity and making the equation more manageable moving forward.

Factoring

Factoring is an essential technique in algebra that involves breaking down expressions into simpler terms or products of their factors. For the given equation: \[ S = 2 \pi r h + 2 \pi r^{2} \] We notice that \(2 \pi r\) is a common factor in both terms on the right-hand side. This allows us to factor it out: \[ S = 2 \pi r(h + r) \]

• Factoring transforms the expression into a product of its factors.
• It simplifies the equation and allows for easier isolation of variables later on.

This step is crucial as it simplifies the equation substantially, setting the stage for the next phase, which is isolating the variable we want to solve for.

Isolating Variables

Once the equation is simplified, the next step is to isolate the variable we want to solve for. This means manipulating the equation so that the variable is on one side of the equation and everything else is on the other. For our equation: \[ S = 2 \pi r(h + r) \] We aim to isolate \(r\). To do this, we divide both sides of the equation by \(2 \pi (h + r)\): \[ \frac{S}{2 \pi (h + r)} = r \] This step puts \(r\) alone on one side of the equation and everything else on the other. It's the final step to solving the equation for \(r\).

• Isolating variables involves using inverse operations to 'undo' the operations around the variable.
• This is achieved through division, multiplication, addition, or subtraction.

By performing these steps, we can isolate any variable and solve for it efficiently, ensuring we find the correct solution.