Problem 10 - Contemporary Precalculus

Question

Solve the equation for the indicated variable. $$V=\pi b^{2} c \quad \text { for } c$$

Step-by-Step Solution

Answer

Question: Given the equation for the volume of a cone, V = πb²c, where V is the volume, b is the base radius, and c is the height, solve the equation for the height variable, c. Solution: c = 𝑉 / (πb²)

Step 1: Isolate the term with the variable

First, we need to isolate the term containing the variable \(c\) on one side of the equation. In this case, it is already isolated on the right side, so we can move to the next step.

Step 2: Divide both sides of the equation by the coefficient of the variable

To isolate the variable \(c\), we need to divide both sides of the equation by the coefficient of \(c\), which in this case is \(\pi b^2\). This will give us the value of the variable \(c\). $$\frac{V}{\pi b^2} = \frac{\pi b^2 c}{\pi b^2}$$

Step 3: Simplify the equation

Now, cancel out the common terms in the equation on the right side. In this case, the common terms are \(\pi b^2\). $$\frac{V}{\pi b^2} = \frac{\cancel{\pi b^2} c}{\cancel{\pi b^2}} \Rightarrow \frac{V}{\pi b^2} = c$$

Step 4: Write the final solution

The isolated variable \(c\) is now found, and the equation has been simplified. Write down the final solution as: $$c = \frac{V}{\pi b^2}$$

Key Concepts

Isolate Variable

Before we delve into the intricacies of solving for a specific variable, let's take a moment to understand why isolating the variable is so important. In algebra, when we are asked to solve for a variable, we are often dealing with an equation that has multiple terms and variables mixed together. To find the value of one specific variable, our goal is to 'isolate' that variable on one side of the equation, leaving a simple statement of equality between the variable and a numerical expression or another variable.

In the given exercise, solving for the variable 'c' essentially requires that we manipulate the equation so that 'c' stands alone on one side. This process typically involves undoing operations that are applied to the variable, like addition, subtraction, multiplication, or division. Once isolated, the variable 'c' can be easily identified with its corresponding value or expression, making it crystal clear to anyone who looks at the solution.

Algebraic Manipulation

Algebraic manipulation is akin to a balancing act. Just as a tightrope walker maintains balance, we must keep our equation balanced while we perform operations to isolate our variable. This process involves a series of techniques and moves, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression.

Operation Techniques

When we have an equation like the one in our exercise, \(V = \pi b^2 c\), our goal is to untangle \(c\). But we want to do so in a way that maintains the integrity of the equation. To this end, we do the following:

• Identify the operations affecting the variable.
• Perform the inverse operation to both sides to 'cancel' the effect on the variable.
• Keep the equation balanced by treating both sides equally.

In the solution, dividing both sides by \(\pi b^2\) ensures that we do not disrupt the balance of the equation while we free \(c\) from its coefficient.

Equation Simplification

Now that we've performed the necessary algebraic manipulation, it's time to clean things up—a process known as equation simplification. Simplifying an equation is about making it as straightforward as possible, often by cancelling out terms or reducing fractions. This is not just about aesthetics; a simplified equation makes it easier to understand and to use for subsequent calculations.

Cancel Out the Common Terms

In our equation, we see that by dividing by \(\pi b^2\), we simplify the equation by cancelling out the terms on the right side. Simplification is like tidying up our mathematical workspace; we remove clutter, highlight what's important, and ultimately, result in an equation that is easy to read and interpret, \(c = \frac{V}{\pi b^2}\). This final form is like a mathematical beacon, showing the way to the value of 'c' clearly and without distraction.