Problem 62 - College Algebra

Question

In Exercises \(55-74\), solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$ V-\pi r^{2} h \text { for } h $$

Step-by-Step Solution

Answer

The isolated formula for h is \(h=\frac{V}{\pi r^{2}}\)

Step 1: Identify Formula's Meaning

Identify the formula as the volume of a cylindrical shape. Here, 'V' represents volume, 'h' is height, and 'r' is radius.

Step 2: Solve for Specified Variable

The goal is to solve the equation for 'h'. This can be done by dividing both sides of the formula by \(\pi r^{2}\). This will isolate 'h' on one side of the equation.

Step 3: Presentation of Solution

Applying the isolation of 'h' from the previous step and present the solution, this generates the formula \(h=\frac{V}{\pi r^{2}}\).

Key Concepts

Cylinder Volume Formula

To tackle problems related to the volume of a cylinder, it's crucial to understand the formula first. The formula to calculate the volume of a cylinder is given by: \[ V = \pi r^2 h \] where:

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

This formula calculates the volume by taking the area of the base, which is a circle (\( \pi r^2 \)), and multiplying it by the height of the cylinder. Understanding each element of this formula helps simplify many geometry and physical problems involving cylindrical shapes.

Isolating Variables

In algebra, isolating a variable means rearranging an equation such that one specific variable is alone on one side of the equation. This process is crucial for finding specific measurements, like height in a cylinder volume formula. Let's look at how to isolate a variable, using our example problem:
To isolate \( h \) in the cylinder volume formula:

• Start with the formula \( V = \pi r^2 h \).
• The goal is to get \( h \) by itself on one side of the equation.
• Divide both sides of the equation by \( \pi r^2 \) to solve for \( h \).
• This gives us \( h = \frac{V}{\pi r^2} \), effectively isolating \( h \).

This process is a fundamental algebraic skill, useful in a range of math problems.

Algebraic Manipulation

Algebraic manipulation is the technique of using mathematical operations to simplify or rearrange equations. It often involves performing operations on both sides of an equation to maintain equality. Let's break down this process further:

• **Identifying operations**: Determine what operations have been applied to the variable you want to isolate. For the volume formula, \( h \) is multiplied by \( \pi r^2 \).
• **Undoing operations**: Apply the inverse operation to both sides. Since \( h \) is multiplied, you'll divide. This maintains balance in the equation.
• **Performing the operation**: Divide both sides by \( \pi r^2 \). It's crucial here that you apply this division to the whole side, not just part of it.
• **Simplifying**: Once the variable is isolated, you should have \( h = \frac{V}{\pi r^2} \).

The ability to manipulate and solve equations is essential across numerous disciplines, not just in mathematics. It helps in converting complex problems to simpler ones, aiding in intuitive understanding and problem-solving.