Problem 19 - Algebra for College Students

Question

$$\text { Solve each formula for the indicated variable.}$$ $$V=\frac{4}{3} \pi r^{2} h \text { for } h$$

Step-by-Step Solution

Answer

The solution for \(h\) is \( h = \frac{3V}{4 \pi r^2} \).

Step 1: Understand the Formula

We are given the volume formula for a cylinder: \[ V = \frac{4}{3} \pi r^2 h \] We need to solve for the variable \(h\).

Step 2: Isolate h

To solve for \(h\), we need to get \(h\) by itself on one side of the equation. We start by eliminating the fraction and other terms on the right-hand side. Divide both sides by \(\frac{4}{3} \pi r^2\): \[ h = \frac{V}{\frac{4}{3} \pi r^2} \]

Step 3: Simplify the Equation

Simplify the right-hand side of the equation by multiplying both the numerator and the denominator by the reciprocal of \(\frac{4}{3}\): \[ h = \frac{V \cdot 3}{4 \pi r^2} \] This simplifies to: \[ h = \frac{3V}{4 \pi r^2} \]

Key Concepts

Isolation of Variables

When solving equations, one common strategy is isolating a variable. This means we manipulate the equation to get the variable by itself on one side. In our exercise, we are isolating the variable \(h\) in the formula for the volume of a cylinder. To do this, we start by eliminating other terms on the right-hand side of the equation. We do this by performing the inverse operations. Here’s the step-by-step breakdown:

• Identify the term containing the variable you need to isolate. In our case, it's \(h\).
• Eliminate constants and coefficients by performing operations like division or subtraction on both sides of the equation.
• Keep performing these steps until the variable is alone on one side.

This process helps us rearrange the formula to solve for the required variable efficiently. In our problem, we divided both sides by \(\frac{4}{3} \pi r^2\) and eventually got:
\[ h = \frac{3V}{4 \pi r^2} \]

Formula Manipulation

Formula manipulation is all about rearranging formulas to solve for a specific variable. This often involves a combination of basic algebraic operations such as:

• Multiplication
• Division
• Addition
• Subtraction

In our example, we wanted to isolate \(h\) in the cylinder volume formula:
\[ V = \frac{4}{3} \pi r^2 h \] We needed to move all other terms (\( \frac{4}{3}, \pi, r^2 \)) away from \(h\). Dividing both sides by \( \frac{4}{3} \pi r^2 \) was our first step. Then, we further simplified by multiplying the numerator and the denominator by the reciprocal of \(\frac{4}{3}\). This resulted in:
\[ h = \frac{3V}{4 \pi r^2} \] Formula manipulation is a powerful tool that allows you to derive values for different variables from established equations. It's particularly useful in geometry and physics problems.

Geometry

Geometry often involves using and manipulating different formulas to find measurements such as volume, area, or length. In our example, we dealt with the volume of a cylinder. Here, knowledge of geometric formulas is key. The formula for the volume of a cylinder is:
\[ V = \frac{4}{3} \pi r^2 h \] This standard formula incorporates the radius (\(r\)), the height (\(h\)) and the mathematical constant \(\pi\). In geometric problems, it’s common to be tasked with solving for one missing measurement given certain known values. Knowing how to rearrange and manipulate these formulas, as we did to isolate \(h\), makes solving these problems feasible. Understanding the underlying geometry principles and how to manipulate the formulas is crucial for finding the right solutions.