Problem 39 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ V=\frac{1}{3} \pi r^{2} h, \text { for } h $$

Step-by-Step Solution

Answer

\( h = \frac{3V}{\pi r^{2}} \)

Step 1: Understand the given formula

We are given the formula for the volume of a cone: \( V = \frac{1}{3} \pi r^{2} h \). Our task is to solve this formula for the variable \( h \). This means we need to rearrange the formula so that \( h \) is isolated on one side of the equation.

Step 2: Eliminate the fraction

To get rid of the fraction, multiply both sides of the equation by 3. This gives us: \( 3V = \pi r^{2} h \). By multiplying by 3, we have cleared the fraction, making it easier to solve for \( h \).

Step 3: Isolate \( h \)

Now, to solve for \( h \), divide both sides of the equation by \( \pi r^{2} \). This results in: \( h = \frac{3V}{\pi r^{2}} \). We have now isolated \( h \) on one side, and the expression for \( h \) is complete.

Key Concepts

Isolating Variables

Isolating variables is a crucial skill in algebra and is often necessary for solving equations or rearranging formulas. The process involves rearranging the equation so that the variable of interest stands alone on one side of the equation, typically on the left. Why do we do this? To understand the relationship of one variable in terms of the others.

The steps to isolate a variable are straightforward:

• Identify the term with the variable you want to isolate.
• Use inverse operations to move all other terms to the opposite side of the equation.
• Simplify the equation if needed.

In our example, we wanted to isolate the variable \( h \). We used inverse operations like multiplying and dividing to clear any fractions or coefficients influencing \( h \). This process is essential for solving for unknowns and understanding how factors like radius and volume influence height in geometric formulas.

Volume of a Cone

The volume of a cone is calculated using the formula \( V = \frac{1}{3} \, \pi \, r^2 \, h \). Understanding this formula helps in calculating how much space is inside a cone. Let's break it down:

• \( \frac{1}{3} \): Represents that a cone is a third of the volume of a cylinder with the same base and height.
• \( \pi \): A mathematical constant approximately equal to 3.14159, crucial for calculations involving circles.
• \( r^2 \): This is the radius of the base squared. Squaring the radius is necessary because a cone's base is circular.
• \( h \): The height of the cone, measured from the base to the peak.

This formula allows us to compute the volume efficiently by plugging in the cone's base radius and its height. Once we understood the formula and our goal (isolating \( h \)), rearranging became intuitive.

Rearranging Formulas

Rearranging formulas is about changing the structure of an equation to express it differently. Often, this process involves manipulating the equation using algebraic techniques. Rearranging allows us to highlight a specific variable's role or solve for it directly.

For the formula \( V = \frac{1}{3} \pi r^2 h \), solving for a particular variable (such as \( h \)) lets us know how the height affects the cone's volume with given dimensions. Steps include:

• Identifying the variable to solve for.
• Using arithmetic operations to isolate the variable.
• Simplifying until the variable is expressed in terms of the other variables.

By focusing on the structure, rearranging also aids in understanding relationships amongst quantities and can be applied to any mathematical or scientific equation multiple times as needed.