Problem 87 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The height \(h\) of the cylinder can be calculated using the formula \(h = \frac{V}{\pi r^{2}}\)

Step 1: Volume of a Cylinder

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

Step 2: Isolate the Variable

We want to solve for \(h\), so isolate \(h\) by dividing both sides of the equation by \(\pi r^{2}\). Thus, \(h = \frac{V}{\pi r^{2}}\).

Key Concepts

Solving for Variables

Understanding how to isolate a variable is essential when dealing with algebraic equations. This process involves performing operations that 'undo' what has been done to the variable you're solving for. In the context of volume calculation for a right circular cylinder, we have the equation \(V = \pi r^{2} h\), where \(V\) represents the volume, \(r\) is the radius, and \(h\) represents the height of the cylinder. To solve for \(h\), one must manipulate the equation in such a way that \(h\) stands alone on one side of the equal sign.
The steps to isolate \(h\) are as follows:

• Identify that \(\pi r^{2}\) is being multiplied by \(h\).
• Perform the opposite operation on both sides of the equation to maintain balance. Since \(h\) is multiplied by \(\pi r^{2}\), we will divide both sides by \(\pi r^{2}\).
• The equation then becomes \(h = \frac{V}{\pi r^{2}}\), which shows \(h\) isolated and gives us the formula to find the height of a cylinder when the volume and radius are known.

Right Circular Cylinder

A right circular cylinder is a solid geometric shape with straight parallel sides and a circular base. To envision this, think of a common soup can. The 'right' part of its name indicates that the sides of the cylinder are perpendicular to its base, creating right angles. This shape's uniform cross-section is key to its volume formula.

Components of a Right Circular Cylinder

Every right circular cylinder is determined by two main measurements: the radius (\(r\)) of its circular base and the height (\(h\)) of the cylinder. These dimensions are necessary and sufficient to calculate its volume, surface area, and other properties relevant to the shape. In the case of the textbook exercise, we are using these measurements to find a particular dimension, the height, when provided with the volume and radius.

Geometric Formulas

Geometric formulas are the recipes we use to calculate various properties of shapes and objects. These formulas range from finding the area of a square (\(A = s^2\), where \(s\) is the length of a side) to calculating the volume of a sphere (\(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius). The formula for the volume of a right circular cylinder, \(V = \pi r^{2} h\), falls among these valuable tools.

Why Geometric Formulas Matter

Having a command of geometric formulas empowers students to solve practical problems, from designing objects to calculating materials needed for construction. In addition, understanding the volume formula for a cylinder can lead to solving more complex real-world challenges, such as determining how much water a tank can hold or the amount of concrete needed for a cylindrical pillar.