Problem 72 - Elementary Algebra

Question

For the following problems, solve each literal equation for the designated letter. \(V=\frac{1}{6} \pi\left(3 a^{2}+h^{2}\right)\) for \(h^{2}\)

Step-by-Step Solution

Answer

Question: Solve the equation \(V=\frac{1}{6} \pi\left(3 a^{2}+h^{2}\right)\) for \(h^{2}\). Answer: \(h^2 = \frac{6V}{\pi} - 3a^2\)

Step 1: 1. Isolate the term with \(h^2\) in it

To do this, first, we will start by multiplying both sides of the equation by 6 to remove the fraction. This gives us: \(6V = \pi(3a^2 + h^2)\)

Step 2: 2. Divide both sides by \(\pi\)

Now, we will divide both sides of the equation by π to further isolate the term with \(h^2\): \(\frac{6V}{\pi} = 3a^2 + h^2\)

Step 3: 3. Subtract \(3a^2\) from both sides

Lastly, we will subtract \(3a^2\) from both sides of the equation to completely isolate \(h^2\): \(\frac{6V}{\pi} - 3a^2 = h^2\) The equation is now solved for \(h^2\).

Key Concepts

Solving for a Variable

In mathematics, solving for a variable in a literal equation involves finding the expression that represents a specific variable within a formula containing one or many variables. Literal equations might look overwhelming at first, especially since they often contain several variables unlike regular equations. The key to mastering them is understanding how to rearrange the components to isolate the variable of interest.
To solve an equation for a variable means to manipulate the equation so that it "looks like" the variable equals something else. For example, solving for \( h^2 \) in the equation \( V = \frac{1}{6} \pi (3a^2 + h^2) \) is a classic example of how these equations function. Each algebraic move must maintain the equality of the equation.
Solving these can be broken down into basic operations like addition, subtraction, multiplication, and division while keeping the equation balanced. This ensures that we properly rearrange the equation without altering the values and expressions involved.

Algebraic Manipulation

Algebraic manipulation refers to the process of using mathematical operations to rearrange equations for various purposes, such as solving for a variable. By employing basic arithmetic, we can alter the form of an equation to make it more useful or understandable.
In the given problem, the presence of a fraction initially complicates direct simplification, but it is effectively handled by multiplying both sides by a factor that eliminates the fraction. Specifically:

• Multiply both sides by 6, dealing with the factor outside the parentheses, when we have \( 6V = \pi (3a^2 + h^2) \).
• This step simplifies the equation and prepares it for further manipulation.

Next, division is used to help further simplify, isolating terms involving the desired variable. This is achieved by dividing both sides by \( \pi \), resulting in \( \frac{6V}{\pi} = 3a^2 + h^2 \). By manipulating through these processes, the structure of the equation changes, allowing for easier access to isolate the desired variable.

Isolation of Variables

Isolation of a variable means rearranging the equation so that the variable appears alone on one side of the equation. This is a critical part of solving literal equations and is particularly useful in formulas across mathematics and science.
Throughout the isolation process, it's essential to:

• Consistently apply inverse operations to "move" other terms in the equation.
• Maintain the equation's balance by performing identical operations on both sides.

In our example, once the term with \( h^2 \) has been isolated to one side—starting from \( \frac{6V}{\pi} = 3a^2 + h^2 \)—the key to achieving complete isolation was subtracting \( 3a^2 \) on both sides: \( \frac{6V}{\pi} - 3a^2 = h^2 \). This leaves \( h^2 \) on its own, showing clearly how each step involved carefully considering how to reduce the equation to a form where \( h^2 \) stands alone.