Problem 19 - Algebra for College Students

Question

$$ V=\pi r^{2} h \quad \text { for } h $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula

We are given the formula for the volume of a cylinder, which is \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, \( h \) is the height of the cylinder, and \( \pi \) is a constant (approximately 3.14159). The task is to solve for \( h \).

Step 2: Isolate the Variable \( h \)

To solve for \( h \), we need to isolate it on one side of the equation. Start with the formula \( V = \pi r^2 h \) and divide both sides by \( \pi r^2 \) to get:\[h = \frac{V}{\pi r^2}\]

Step 3: Verify the Solution

Ensure that the solution correctly solves for \( h \) by substituting it back into the original equation. If \( h = \frac{V}{\pi r^2} \), substituting gives \( V = \pi r^2 \left(\frac{V}{\pi r^2}\right) = V \), which is correct.

Key Concepts

Understanding the Cylinder Volume Formula

The volume of a cylinder can be calculated using the formula \( V = \pi r^2 h \). This formula describes how much space is inside the cylinder. Let's break down the components of this formula:

• \( V \) stands for volume, which is the measure of space within the cylinder.
• \( \pi \) (Pi) is a mathematical constant, approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.
• \( r \) is the radius of the cylinder's circular base. The radius is the distance from the center of the circle to its edge.
• \( h \) is the height of the cylinder, which is the perpendicular distance between the two circular bases.

Multiplying \( \pi \) by the square of the radius \( r^2 \) gives the area of the circular base. When you multiply this area by the height \( h \), you get the total volume of the cylinder. This formula is very handy in geometry when you need to calculate how much liquid a cylindrical tank can hold or the inner capacity of any cylindrical object.

Isolation of Variables in Equations

Isolation of variables is a fundamental skill in algebra that involves solving for one variable in terms of others. In the context of the cylinder's volume formula, we start with \( V = \pi r^2 h \) and aim to solve for \( h \). This means we want to have the height \( h \) by itself on one side of the equation.
To do this, you need to perform algebraic manipulations to both sides of the equation. To isolate \( h \):

• Divide both sides of the equation by \( \pi r^2 \). This operation cancels these terms on the right side of the equation.
• The result is \( h = \frac{V}{\pi r^2} \).

By isolating \( h \), we can quickly determine the height of a cylinder if we know its volume \( V \) and radius \( r \). It's an essential strategy used across different branches of mathematics and science to simplify expressions and solve real-world problems.

Linking Geometry and Algebra

Geometry and algebra often work hand-in-hand, especially when it comes to solving problems involving shapes. The formula \( V = \pi r^2 h \) is an excellent example of how geometric concepts are expressed and manipulated using algebraic equations.
Algebra allows us to take geometric rules and express them in a format that can be manipulated and solved mathematically. When dealing with a cylinder:

• The concepts of radius and height are geometric; they define the shape and size of the cylinder.
• Using algebra to manipulate these terms helps in understanding relationships between volume, radius, and height.
• This linkage provides powerful methods for calculating dimensions across various contexts—whether in architecture, engineering, or physics.

Understanding how to handle such algebraic-geometric relationships is crucial, allowing us to solve for unknowns and make predictions based on given data.