Problem 49 - College Algebra

Question

Solve for \(h : V=\pi r^{2} h\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The formula given is the volume of a cylinder, where \(V\) is the volume, \(r\) is the radius of the base, and \(h\) is the height of the cylinder. Our task is to solve for \(h\), meaning we need to express \(h\) in terms of \(V\) and \(r\).

Step 2: Isolate the Variable

Our goal is to isolate \(h\) on one side of the equation. Start with the equation:\[V = \pi r^2 h\]To isolate \(h\), divide both sides of the equation by \(\pi r^2\):\[\frac{V}{\pi r^2} = h\]

Step 3: Simplify the Equation

The equation \(h = \frac{V}{\pi r^2}\) can be considered as the simplest form as it directly provides \(h\) in terms of the volume \(V\) and the radius \(r\). The height \(h\) is thus calculated by dividing the volume \(V\) by the area of the base, \(\pi r^2\), of the cylinder.

Key Concepts

Volume of a Cylinder

The volume of a cylinder is a measure of how much space is inside the cylinder. It is calculated using the formula:

• \( V = \pi r^2 h \)

where:

• \( V \) represents the volume
• \( \pi \) is a constant approximately equal to 3.14159
• \( r \) is the radius of the circular base
• \( h \) is the height of the cylinder

This formula works because the area of the base \( \pi r^2 \) is multiplied by the height \( h \), which in essence just stacks the area over and over until reaching the height \( h \). Understanding this formula is crucial for solving any problems involving the volume of a cylinder, as it forms the basis for manipulating and isolating necessary variables.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging equations to isolate a particular variable or to simplify expressions. It involves applying mathematical operations like addition, subtraction, multiplication, or division to both sides of an equation to maintain equality.
In our original problem, we start with the equation for the volume of a cylinder:

• \( V = \pi r^2 h \)

The aim is to express one variable (here, \( h \)) in terms of the other known quantities \( V \) and \( r \). This can be done by rearranging the equation, which involves the key step of division in this context.
By dividing both sides by \( \pi r^2 \), we isolate \( h \) and derive the expression:

• \( h = \frac{V}{\pi r^2} \)

Such manipulation is fundamental in algebra as it allows you to focus on any single variable, depending on the information you are given or required to find.

Isolating Variables

Isolating variables is a fundamental technique used to solve equations. The primary goal is to rearrange the equation so that one side contains only the variable you want to find.
This is especially useful in formulas with multiple variables involved, like the volume of a cylinder.
To isolate a variable successfully, you must perform the same operation to both sides of the equation. This keeps the equation balanced. For the equation:

• \( V = \pi r^2 h \)

we isolated \( h \) by dividing both sides by \( \pi r^2 \):

• \( h = \frac{V}{\pi r^2} \)

This step is vital because it expresses \( h \) purely in terms of known quantities (volume \( V \) and radius \( r \)), making it easier to calculate for any given set of values. Always remember, isolating variables is about using inverse operations (like division being the inverse of multiplication) to "undo" what is being done to the variable.