Problem 13 - Algebra 1

Question

Solve for the indicated variable. Volume of a Circular Cylinder Solve for \(h: V=\pi r^{2} h\)

Step-by-Step Solution

Answer

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

Step 1: Understand the formula

The formula given, \(V = \pi r^{2} h\), is the formula for the volume of a circular cylinder, where \(V\) stands for volume, \(\pi\) is a constant that is approximately 3.14159, \(r\) is the radius of the base of the cylinder, and \(h\) is the height of the cylinder.

Step 2: Isolate h

To solve for \(h\), h must be isolated on one side of the equation. This can be done by dividing both sides of the equation by \( \pi r^{2}\). This gives us the equation \(h = \frac{V}{\pi r^{2}}\).

Key Concepts

Solving Equations

Solving equations involves finding the value of an unknown variable that makes the equation true. In the context of a cylinder's volume formula, we start with a given equation:

• The formula for volume: \( V = \pi r^{2} h \)
• Our goal: solve for the height \( h \)

To solve for \( h \), we need to manipulate the equation to express \( h \) in terms of the other variables. This requires isolating \( h \) on one side. By dividing both sides by \( \pi r^2 \), we move everything else over to the other side, yielding \( h = \frac{V}{\pi r^2} \). This is a fundamental skill in algebra, often used to make mathematical models more interpretable. Understanding each step in rearranging the equation can greatly enhance your problem-solving skills.

Formula Manipulation

Formula manipulation is the process of changing a formula to solve for a different variable or parameter. It's a critical skill in mathematics and physics because it allows you to understand the relationships between variables. For the cylinder volume formula \( V = \pi r^{2} h \), manipulating the formula to solve for \( h \) involves division.

• Start by identifying the variable to isolate. Here, it’s \( h \).
• Recognize constants and variables that need to be moved across the equation: \( \pi r^{2} \) which are multiplied with \( h \).
• Use inverse operations, such as division, to isolate \( h \).

By dividing both sides of the equation by \( \pi r^{2} \), we effectively 'move' these terms across the equality sign. The result is the formula \( h = \frac{V}{\pi r^{2}} \), reflecting how height relates inversely with the square of the radius and directly with volume. Mastering formula manipulation deepens understanding and expands its application in complex problems.

Geometry

Geometry, being the study of space and shape, brings about a deeper appreciation for the objects and figures we encounter. A cylinder is one such geometric object. It has a specific formula for its volume: \( V = \pi r^{2} h \).

• A cylinder is defined by its base radius \( r \) and height \( h \).
• The base is a circle, which is why \( \pi r^{2} \) appears in the volume formula.
• Multiplying by height \( h \) extends the area of the circle to 3D space, forming the volume.

Understanding the geometric principles behind formulas helps in visualizing and predicting how changes in dimensions, like radius or height, affect overall volume. Recognizing these principles enriches the learning experience, reinforcing how geometry and algebra interconnect to solve practical problems.