Problem 80 - Elementary Algebra

Question

Solve for the indicated variable. $$ \text { Solve for } h: \quad S=2 \pi r 2+2 \pi r h . $$

Step-by-Step Solution

Answer

\( h = \frac{S - 2 \pi r^2}{2 \pi r} \).

Step 1: Identify the Terms Involving 'h'

The equation given is \( S = 2 \pi r^2 + 2 \pi r h \). We need to solve for \( h \). Identify the term that contains \( h \), which in this case is \( 2 \pi r h \).

Step 2: Isolate the 'h' Term

To isolate the term \( 2 \pi r h \), subtract \( 2 \pi r^2 \) from both sides of the equation. This gives you \( S - 2 \pi r^2 = 2 \pi r h \).

Step 3: Solve for 'h'

Now that the term \( 2 \pi r h \) is isolated, divide both sides of the equation by \( 2 \pi r \) to solve for \( h \): \[ h = \frac{S - 2 \pi r^2}{2 \pi r} \].

Key Concepts

Solving Equations

Solving equations is a fundamental part of algebra. When you solve an equation, you're looking to find the value of the variable that makes the equation true. In simple terms, an equation is like a balance. Whatever you do to one side must be done to the other.
A typical process to solve equations involves:

• Identifying known and unknown values: Look for numbers and variables. Variables are the unknowns you need to solve for, while numbers or coefficients are known values.
• Manipulating the equation: Use mathematical operations such as addition, subtraction, multiplication, and division to rearrange the equation.
• Maintaining balance: The same operation must be applied to both sides of the equation. This ensures that the equality remains true.

Solving equations is like following a recipe; every step has a purpose in helping you isolate the variable. With practice, solving equations becomes predictable and systematic.

Variables

In algebra, variables are symbols or letters that represent unknown values. For example, in the equation \( S = 2 \pi r^2 + 2 \pi r h \), \( h \) is the variable. Variables allow algebraic expressions to model real-world situations, making them essential in representing unknowns in equations.
Variables behave consistently through rules of operations, just like numbers. They can:

• Be combined: If you have multiple instances of the same variable, you can add or subtract them.
• Be multiplied or divided: You can perform these operations with variables to simplify expressions.
• Represent any number: Variables stand for values that can change, which means they hold the possibility of representing various numbers.

Variables make mathematical modeling flexible and powerful, allowing for solutions to unknown situations by manipulating these symbols until we solve for them.

Isolation of Terms

Isolation of terms is a crucial concept in solving equations. It involves getting the unknown variable by itself on one side of the equation. This requires reversing operations in a strategic way. In our example \( S = 2 \pi r^2 + 2 \pi r h \), the goal is to isolate \( h \).
To isolate terms, you can:

• Identify all terms involving the variable: Look at both sides of the equation and recognize terms with the variable you're solving for.
• Undo operations: Perform reverse operations of what's affecting the variable. If the variable is added or subtracted, do the opposite operation.
• Use systematic techniques: Such as factoring, distributing, or using inverses to simplify and isolate terms.

Isolating the variable term is a methodical process. By understanding and applying the right operations, you can simplify even complex equations into a form where the solution for the variable is clear and straightforward.