Problem 38 - Algebra Form and Function

Question

Is the given expression linear in the indicated variable? Assume all constants are non-zero. $$ 2 \pi r^{2}+\pi r h, h $$

Step-by-Step Solution

Answer

Expression: \(2 \pi r^{2} + \pi r h\) Answer: Yes, the expression is linear in the variable h.

Step 1: Identify the components of the expression

Break down the given expression into individual terms: $$ \underbrace{2 \pi r^{2}}_{\text{Term 1}} + \underbrace{\pi r h}_{\text{Term 2}}, $$

Step 2: Find the power of variable h in each term

The given expression has two terms; we will find the power of h in each term: - Term 1: \(2 \pi r^{2}\) - The variable h is not present in this term. - Term 2: \(\pi r h\) - The power of h in this term is 1.

Step 3: Determine the linearity of the expression in variable h

Since the power of variable h is 1 in Term 2 and not present in Term 1, the expression is linear concerning the variable h.

Key Concepts

Understanding Variables

Variables in algebra are symbols used to represent unknown quantities or changing values. They are typically denoted by letters such as \(x\), \(y\), or \(h\).
In our example, the variable is \(h\). Variables help us simplify and solve mathematical problems because they allow us to express general relationships.
In any algebraic expression, variables can have different roles:

• They can stand alone, like \(h\) in \(\pi r h\).
• They can be part of a term, multiplying with constants, such as \(\pi r h\).
• They can be raised to a power, reflecting their influence in the expression.

Observing how variables interact within terms will help determine the expression's complexity and nature.

Exploring Terms in Algebra

Terms are the building blocks of algebraic expressions. A term can be a single number, a variable, or numbers and variables multiplied together. In the expression \(2 \pi r^{2} + \pi r h\), there are two terms:

• The first term is \(2 \pi r^{2}\).
• The second term is \(\pi r h\).

Each term is separated by addition or subtraction, signaling a new part of the expression. It's crucial to identify each term properly because they define what we're working within an equation.
By examining terms closely, such as identifying coefficients and variables, we can easily determine properties like linearity or polynomial degree.

Powers of Variables in Expressions

The power with which a variable is raised in a term indicates the expression’s degree concerning that variable. In general, a power or exponent signifies how many times a variable is multiplied by itself. For a term to be linear, any variable within must have a power of exactly 1.
In our problem expression, we identify two terms:

• Term 1: \(2 \pi r^{2}\) - The variable \(h\) does not appear, so it is not impactful in this term's linearity with respect to \(h\).
• Term 2: \(\pi r h\) - The power of \(h\) is 1, confirming that this term is linear regarding \(h\).

Thus, we conclude that the expression is linear in terms of \(h\) because the highest power of \(h\) in the expression is 1. Understanding the power of a variable is essential as it directly impacts the function's behavior and its classification, like linear, quadratic, or cubic.