Problem 64 - College Algebra

Question

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? 64\. \(V=\pi r^{2} h\) for \(h\)

Step-by-Step Solution

Answer

The variable h, in terms of V, r and \( \pi\), is \(h = \frac{V}{\pi r^{2}}\).

Step 1: Write down the given formula

The given formula is \(V=\pi r^{2} h\). We are tasked to solve for h, which means we need to express h in terms of V, r and \( \pi\).

Step 2: Isolate the variable

To solve for h, we need to get h by itself on one side of the equation. We can do this by dividing both sides of the equation by \( \pi r^{2}\).

Step 3: Perform the operation

Dividing both sides of the given formula by \( \pi r^{2}\) , we get \(h = \frac{V}{\pi r^{2}}\). This is the expression for h, in terms of V, r and \( \pi\).

Key Concepts

Algebraic Manipulation

Understanding algebraic manipulation is foundational for effectively solving for variables within equations. At its core, algebraic manipulation involves applying mathematical operations to both sides of an equation in order to maintain its equality while transforming its structure. This process is guided by the properties of arithmetic operations such as addition, subtraction, multiplication, and division, as well as the distributive property and the ability to combine like terms.

For instance, taking the formula from the exercise, \(V = \$\pi r^{2} h\), the goal is to express the variable \(h\) in terms of the other variables and constants. Manipulating this equation successfully requires familiarity with these arithmetic operations, a skill that is essential in algebra and beyond. In educational contexts, strengthening your algebraic manipulation skills leads to increased mathematical literacy and a more comprehensive understanding of mathematical concepts as a whole.

Formula Rearrangement

Formula rearrangement is a crucial step in solving for a particular variable. It involves reorganizing the terms of an equation to isolate the variable of interest. This procedure makes it easier to see the relationship between the variable and the other parts of the equation. It is a methodical approach that often starts with simplification or expansion of the equation, followed by moving terms to different sides accordingly.

In the provided exercise, we saw the formula \(V = \$\pi r^{2} h\), which is the volume of a cylinder. To solve for \(h\), formula rearrangement is necessary. The process starts by identifying the terms that are associated with \(h\), which in this case are \(\$\pi r^{2}\). By dividing the entire equation by these terms, we can isolate \(h\) effectively. This strategic shuffling of terms in an equation is what makes the problem solvable, and mastering it can greatly enhance one's problem-solving capabilities.

Isolating Variables

Isolating a variable is the decisive step that completes the process of solving a mathematical equation. It is all about 'setting the variable free' from the other terms in the equation so that it stands alone and its value can be clearly determined in terms of other known quantities. Isolating the variable conveys clarity and allows one to see the direct relationships between different quantities within an equation.

In the case of our cylinder volume equation, \(V = \$\pi r^{2} h\), isolating \(h\) involves dividing both sides of the equation by \(\$\pi r^{2}\). The outcome, \(h = \frac{V}{\$\pi r^{2}}\), does exactly that — \(h\) is now on one side of the equation, with a clear expression indicating how it is calculated from \(V\), \(r\), and \(\$\pi\). This isolated form lays bare the direct proportionality between the height of the cylinder and its volume, for a given radius.