91Ó°ÊÓ

A cone is a three-dimensional geometric shape with a circular base and a single vertex. Understanding the volume of a cone is crucial when working with various geometry problems. The formula to calculate the volume of a cone is:
\(V = \frac{1}{3} \pi r^{2} h \)
Here, \(V\) represents the volume, \(\pi\) is a constant (approximately 3.14159), \(r\) is the radius of the base, and \(h\) is the height.
This formula essentially tells us that the volume of a cone is one-third the volume of a cylinder with the same base and height. Visualize the cone being filled with small slices or disks stacked upon each other—this stacking gives the cone its volume. Familiarizing yourself with this formula can help you quickly navigate geometry problems involving cones.

Algebraic manipulation involves rearranging and simplifying expressions and equations to solve for a particular variable. It's like reshuffling parts of an equation to see relationships more clearly or find unknown values. This skill is vital for solving many types of math problems.
In the given problem, we start with the cone volume formula:
\[ V = \frac{1}{3} \pi r^{2} h \]
Our goal is to solve for the height \(h\). First, we need to get rid of the fraction. We do this by multiplying both sides by 3:
\[ 3V = \pi r^{2} h \]
This step simplifies the equation and makes it easier to work with. Effective algebraic manipulation often involves such steps—clearing fractions or multiplying both sides by the same number to simplify an equation.

Isolating a variable means getting it alone on one side of an equation. This is essential for solving equations and understanding how changes in one variable affect another. In the final step of our problem, we isolate the height \(h\).
From our previous manipulation, we have:
\[ 3V = \pi r^{2} h \]
To isolate \(h\), divide both sides of the equation by \(\pi r^{2}\):
\[ h = \frac{3V}{\pi r^{2}} \]
Now \(h\) is by itself, and we have a clear expression for it in terms of volume \(V\) and radius \(r\).
When practicing isolating variables, always perform the same operation on both sides of the equation. This maintains the balance of the equation and helps you arrive at the correct solution.
Mastering this concept allows you to solve more complex equations and understand relationships between different variables efficiently.