Question

Volume of a Cylinder The volume \(V\) of a right circular cylinder of height \(h\) and radius \(r\) is \(V=\pi r^{2} h .\) If the height is twice the radius, express the volume \(V\) as a function of \(r\)

Short answer

The volume \( V \) as a function of \( r \) is \( V = 2 \pi r^3 \).

Step-by-step solution

- Recall the formula for the volume of a cylinder

The volume of a right circular cylinder can be found using the formula: \[ V = \pi r^2 h \] where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.

- Express height in terms of the radius

According to the problem, the height \( h \) is twice the radius \( r \). This can be written as: \[ h = 2r \]

- Substitute the expression for height into the volume formula

Replace \( h \) in the volume formula with \( 2r \): \[ V = \pi r^2 (2r) \]

- Simplify the expression

Simplify the equation to express the volume as a function of the radius \( r \): \[ V = \pi r^2 (2r) = 2 \pi r^3 \]

Key concepts

right circular cylinder

A right circular cylinder is a three-dimensional geometric shape. It has two parallel circular bases connected by a curved surface. The 'right' part of its name means that the cylinder's height is perpendicular to its bases. This makes measuring and calculating properties like volume and surface area much easier. You'll often see right circular cylinders in everyday objects like soda cans or pipes. These shapes are quite common because they efficiently use space.

volume formula

To find the volume of a right circular cylinder, we use the following formula: \[ V = \pi r^2 h \] Here, \ (r) represents the radius of the base, and \ (h) stands for the height of the cylinder. Calculating volume is straightforward. You square the radius (multiply it by itself) and then multiply this area value by the height. Finally, you multiply by \ \pi (approximately 3.14159). This formula gives you the space inside the cylinder, measured in cubic units. Always remember to keep units consistent—for example, using inches or cm for both radius and height.

express in terms of radius

In some problems, you'll need to express volume using only one variable—in this case, the radius. Going back to our exercise, we know the height \ (h) is twice the radius \ (r). This turns our formula into: \[ h = 2r \] Substituting \ (h) with \ (2r) in the original volume formula: \[ V = \pi r^2 (2r) \] Simplifying, we get: \[ V = 2 \pi r^3 \] Now, the volume is written entirely in terms of the radius. This form helps in many practical applications because it reduces the number of variables you need to measure or estimate.