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To find the volume of a cylinder, you need to understand the volume formula. A right circular cylinder is a 3D shape with two parallel circular bases and a curved surface connecting them. The formula for finding the volume of a right circular cylinder is given by:

\( V = \pi r^2 h \)

Here, V denotes the volume, \(r\) is the radius of the circular base, and \(h\) is the height of the cylinder. By using this formula, you can calculate the amount of space (volume) inside the cylinder.

It's important to have all the values or be able to solve for one of them if the other two are given. In our exercise, we are given the volume \( V = 144\pi \) cubic inches and the radius \( r = 6 \) inches.

Once you have the volume formula at hand, the next step is to substitute the known values. In the problem, the radius \( r \) of the cylinder is given as 6 inches. We substitute this value into the volume formula:

\( 144\pi = \pi (6)^2 h \)

Substitution helps us transform the formula into a simpler equation that we can work with. Here, we replace \( r \) with 6, giving us:

\( 144\pi = \pi \cdot 36 \cdot h \)

This is because \( 6^2 = 36 \). Substituting values is a crucial step in mathematical problem-solving as it helps simplify the problem, making it easier to solve.

With the simplified equation, our goal is to find the height \( h \) of the cylinder. We now focus on isolating \( h \). From the earlier step, we have:

\( 144\pi = \pi \cdot 36 \cdot h \)

To solve for \( h \), divide both sides of the equation by 36\( \pi \):

\( h = \frac{144\pi}{36\pi} \)

The \( \pi \) terms on both the numerator and the denominator cancel out, simplifying to:

\( h = \frac{144}{36} \)

Which further simplifies to:

\( h = 4 \)

This step-by-step process shows that the height of the cylinder is 4 inches. Dividing and simplifying the equation are crucial steps in finding the unknown variable.

Understanding what a right circular cylinder is can help grasp the geometry involved. A right circular cylinder consists of two parallel circular bases connected by a curved surface perpendicular to the bases' plane.

The term 'right' means the sides (or the curved surface) are perpendicular to the base, forming a 90-degree angle. This is crucial for applying the volume formula.

Key characteristics of a right circular cylinder:

• Two identical circular bases
• A perpendicular height \( h \) connecting the bases
• The radius \( r \) of the base circles is constant

With these basic properties clear, you'll find it easier to apply the volume formula and solve problems involving right circular cylinders.