Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve for the indicated variable. Volume of a Right Circular Cone $$\text { Solve for } h: \quad V=\frac{1}{3} \pi r^{2} h$$

Step-by-Step Solution

Answer

The height \(h\) of the cone, given its volume \(V\) and radius \(r\), can be calculated with the formula \(h = \frac{3V}{\pi r^{2}}\).

Step 1: Identify the target variable

The objective is to isolate the variable \(h\) in the formula \(V=\frac{1}{3} \pi r^{2} h\). This means you need to manipulate the equation to make \(h\) the subject.

Step 2: Isolate the variable \(h\)

To get \(h\) alone on one side, divide both sides of the equation by \(\frac{1}{3} \pi r^{2}\). Doing this, the equation becomes \(h = \frac{3V}{\pi r^{2}}\).

Step 3: Simplify the equation

There are no further simplifications possible. The equation in terms of \(h\) is \(h = \frac{3V}{\pi r^{2}}\).

Key Concepts

Volume of a Right Circular Cone

Understanding the volume of a right circular cone is essential when diving into problems involving geometry and algebra. The formula for the volume of a right circular cone is expressed as \( V = \frac{1}{3} \pi r^2 h \). This equation calculates the space occupied within the cone, where:

• \( V \) is the volume
• \( r \) is the radius of the base
• \( h \) is the height

The cone's volume accounts for only a portion of the volume of a cylinder with an equal base and height since it is literally one-third of this value. The factor \( \frac{1}{3} \) in the formula highlights this reduction. By understanding this concept, you can visualize how the cone integrates into broader geometric contexts.

Isolating Variables

In algebra, isolating the variable is a critical skill that allows you to solve for any given unknown in equations. Isolating a variable means to get it alone on one side of the equation. For the formula \( V = \frac{1}{3} \pi r^2 h \), we need to "solve for \( h \)" means we manipulate the expression so that \( h \) is by itself.
To do this effectively:

• Identify the variable to isolate: Here, \( h \).
• Perform inverse operations to remove other terms: You divide both sides by any term that includes \( h \) in the denominator to "free" \( h \).
• Keep any operations balanced: Ensure you perform the same operation to both sides.

In this problem, isolating \( h \), requires acknowledging the division involved and applying it to the entire equation, leading to the solution \( h = \frac{3V}{\pi r^2} \). This step is foundational for solving any algebraic equation.

Solving Equations

Solving equations is all about finding the values of unknown variables that make the equation true. To solve the equation \( V = \frac{1}{3} \pi r^2 h \) for the height \( h \), understanding simplification techniques and logical reasoning is important.
Steps in solving this equation involve:

• Identifying the mathematical operations involved: Multiplication by a fraction (\( \frac{1}{3} \pi r^2 \)) is the key here.
• Reversing these operations: Division is used to counteract multiplication and isolate \( h \).
• Maintaining equality: Whatever operation is applied to one side must be applied to the other, ensuring balance.

The solved equation \( h = \frac{3V}{\pi r^2} \) showcases how algebra transforms expressions into useful forms. By rearranging terms methodically, you uncover the specific value or relationship you're investigating. This process of solving equips you with tools to approach a range of mathematical and real-world problems confidently.