Problem 16 - Algebra for College Students

Question

Solve each formula for the specified variable. \(V=\frac{1}{3} \pi r^{2} h\) for \(h \quad\) (volume of a cone)

Step-by-Step Solution

Answer

The formula solved for \(h\) is: \[h = \frac{3V}{\pi r^{2}}\]

Step 1: Understand the formula

The formula provided is the volume of a cone: \[V = \frac{1}{3} \pi r^{2} h\] We need to solve this formula for the variable \(h\).

Step 2: Isolate the variable h

To isolate \(h\), multiply both sides of the equation by 3 to get rid of the fraction: \[3V = \pi r^{2} h\]

Step 3: Divide by the constants

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

Step 4: Simplify the equation

The formula is now simplified and \(h\) is isolated: \[h = \frac{3V}{\pi r^{2}}\].

Key Concepts

Isolating Variables

Isolating a variable means getting the variable you are solving for by itself on one side of the equation. This process involves moving other terms to the opposite side of the equation through operations such as addition, subtraction, multiplication, or division. In our exercise, we needed to isolate the variable \(h\) in the volume of a cone formula. The formula given was \[V = \frac{1}{3} \pi r^{2} h\].
To start isolating \(h\), we first need to multiply both sides by 3 to remove the fraction:
\[3V = \pi r^{2} h\].
Next, we divide both sides by \(\pi r^{2}\) to completely isolate \(h\):
\[h = \frac{3V}{\pi r^{2}}\].
By isolating variables, we can solve for unknowns effectively.

Volume of a Cone

The volume of a cone is calculated using the formula: \[ V = \frac{1}{3} \pi r^{2} h \].
This formula helps determine the space a cone occupies. The variables are:

• \(V\): Volume of the cone
• \(r\): Radius of the base of the cone
• \(h\): Height of the cone

The fraction \(\frac{1}{3}\) accounts for the cone tapering to a point, unlike a cylinder which uses \(\pi r^{2} h\). By understanding this volume formula, we can perform algebraic manipulation to isolate any variable, just like we did to find \(h\).

Algebraic Manipulation

Algebraic manipulation involves reorganizing equations to isolate a specific variable. This skill is crucial for solving complex mathematical problems. Here's how we apply algebraic manipulation:

• Identify the term containing the variable to be isolated.
• Use inverse operations to move constants and coefficients to the other side of the equation.
• Simplify to isolate the variable fully.

In our problem, we manipulated \[V = \frac{1}{3} \pi r^{2} h\] by:

• Multiplying both sides by 3
• Dividing both sides by \(\pi r^{2}\)

Each step made \(h\) the subject of the formula, simplifying the solution to \[h = \frac{3V}{\pi r^{2}}\].

Geometry Formulas

Geometry involves many formulas to calculate areas, volumes, and other properties of shapes. Knowing how to use and manipulate these formulas is crucial for solving geometric problems accurately. The volume of a cone formula, \[V = \frac{1}{3} \pi r^{2} h\], is just one example. Important related formulas include:

• Volume of a cylinder: \(\text{Volume} = \pi r^{2} h\)
• Area of a circle: \(\text{Area} = \pi r^{2}\)
• Surface area of a sphere: \(\text{Surface Area} = 4 \pi r^{2}\)

Understanding these formulas and how to manipulate them helps in solving various geometric problems, like finding the volume or height of shapes.