Problem 62 - College Algebra Essentials

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$V=\pi r^{2} h \text { for } h$$

Step-by-Step Solution

Answer

The formula \( h = \frac{V}{\pi r^{2}} \) is the result of solving the volume formula of a cylinder for \( h \).

Step 1: Identify the formula

The given formula, \( V = \pi r^{2}h \), describes the volume of a cylinder where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height.

Step 2: Re-arrange the formula to solve for \( h \)

To isolate \( h \), we need to perform algebraic operation on the equation. The process required is the division of the both sides of the equation by \( \pi r^{2} \) to clear out \( \pi r^{2} \) from the right side of the equation.

Step 3: Perform the operation and simplify

Dividing both sides by \( \pi r^{2} \), we get \( h = \frac{V}{\pi r^{2}} \) which is the formula for \( h \).

Key Concepts

Cylinder Volume Formula

The cylinder volume formula is a fundamental expression in geometry and is key to various applications in mathematics and physics. The formula \( V = \pi r^2 h \) defines the volume (\( V \)) of a cylinder as the product of \( \pi \) (pi, approximately 3.14159), the radius of the base squared (\( r^2 \)), and the height (\( h \)) of the cylinder. Recognizing this formula is crucial for solving problems involving the volume of cylindrical objects, such as tanks, pipes, and even some everyday items like cans.
When dealing with problems, identifying the formula not only enables the student to proceed with calculations but also links the abstract numbers to physical concepts, thus enhancing comprehension and retention of information.

Algebraic Manipulation

Algebraic manipulation is the process of transforming an algebraic expression or equation into a desired form using mathematical operations. These operations can include addition, subtraction, multiplication, division, and the application of exponents. In our exercise, algebraic manipulation involves rearranging the cylinder volume formula to solve for a specific variable, in this case, the height (\( h \)). The starting point is understanding what we need to derive and then selectively applying operations that will simplify and isolate the required variable. It's a critical skill in algebra, allowing students to solve for unknowns and reformulate equations to make them more useful for specific applications.

Isolating Variables

Isolating variables is a staple strategy in algebra, particularly useful when solving equations. The goal is to get the variable of interest alone on one side of the equation, which then allows us to find its value. This often requires performing the inverse of whatever operation is currently affecting the variable. For instance, if a variable is being multiplied by a factor, we can isolate it by dividing both sides of the equation by that factor. In our cylinder volume example, to isolate \( h \), we divide both sides of the equation \( V = \pi r^2 h \) by \( \pi r^2 \), giving us \( h = \frac{V}{\pi r^2} \). Learning to isolate variables is not just about solving for them but understanding the balance and symmetry in equations.

Mathematical Operations

Mathematical operations are the building blocks of algebra and encompass addition, subtraction, multiplication, and division. They allow us to manipulate and solve equations. They follow a set of hierarchy and rules, often summarized by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our solution process, we employ division to both sides of the cylinder volume equation to isolate the variable \( h \). Each operation we apply must maintain the balance of the equation, meaning whatever we do to one side must be done to the other, ensuring the equality remains true. Mastering these operations and understanding when and how to apply them is essential for students to effectively analyze and solve a broad range of math problems.