Problem 43 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. $$ V=\frac{1}{3} \pi r^{2} h \quad \text { for } h $$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Variable

In the given formula, we want to solve for the variable \( h \). Begin by multiplying both sides of the equation by \( 3 \) to get rid of the fraction:\[ 3V = \pi r^2 h \].

Step 2: Solve for h

Next, divide both sides of the equation by \( \pi r^2 \) to isolate \( h \):\[ h = \frac{3V}{\pi r^2} \].

Key Concepts

Solving Formulas

Formulas are equations that express relationships between different variables. Solving formulas often involves finding the value of one variable in terms of the others. This is a crucial skill in math and science, enabling you to understand how changes in one quantity affect another.

When solving formulas, begin by identifying the variable you need to solve for. This will guide your steps in rearranging the equation. It's important to keep the equation balanced, meaning whatever operation you apply to one side, you must apply to the other as well.

• Look for operations involving the target variable.
• Perform inverse operations to isolate the target variable.
• Check your work by substituting the solution back into the original equation.

Understanding how to manipulate and solve formulas helps in not just algebra but also in applying mathematical principles to real-world problems, such as calculating areas and volumes in geometry.

Isolating Variables

Isolating variables is an essential part of solving mathematical equations. It involves moving other terms and coefficients away from the variable of interest.

In the equation presented, our aim was to isolate the variable \( h \). We started with a formula for the volume of a cone: \[ V = \frac{1}{3} \pi r^2 h \]To remove the coefficient of \( \frac{1}{3} \), we multiplied both sides by 3. This is a common first step when dealing with fractions:\[ 3V = \pi r^2 h \]The next move was to divide both sides by \( \pi r^2 \) to fully isolate \( h \):\[ h = \frac{3V}{\pi r^2} \]Isolating variables often requires several steps, but by consistently applying algebraic principles, you can clarify the relationships between the variables in any formula.

Geometry Formulas

Geometry involves various formulas to calculate dimensions like area, volume, and surface area. Understanding these formulas provides insights into the properties of shapes and objects.

In the example given, the task was to rearrange the formula for the volume of a cone. Geometry formulas can often seem daunting, but by breaking them down into smaller steps, they are easier to understand.

• The formula involves constants like \( \pi \), which are universal, representing the ratio of the circumference to the diameter of a circle.
• \( r^2 \) shows that volume is directly proportional to the square of the radius.
• The variable \( h \) represents the cone's height, showing its linear effect on volume.

Geometry not only arms you with the tools to solve problems but also helps in visualizing concepts, offering a concrete understanding of abstract mathematical ideas.