Question

Volume of a Cone The volume \(V\) of a right circular cone is \(V=\frac{1}{3} \pi r^{2} h .\) 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 \( V = \frac{2}{3} \pi r^3 \).

Step-by-step solution

Identify the given formula

The formula for the volume of a right circular cone is given by \[ V = \frac{1}{3} \, \pi \, r^2 \, h \]where \(V\) is the volume, \(r\) is the radius of the base, and \(h\) is the height.

Express height in terms of radius

According to the problem, the height \(h\) is twice the radius \(r\). Thus, we can write: \[ h = 2r \]

Substitute height into the volume formula

Replace \(h\) in the volume formula with \(2r\):\[ V = \frac{1}{3} \, \pi \, r^2 \, (2r) \]

Simplify the expression

Simplify the expression to find the volume function in terms of radius \(r\):\[ V = \frac{1}{3} \, \pi \, r^2 \, (2r) = \frac{1}{3} \, \pi \, 2r^3 = \frac{2}{3} \, \pi \, r^3 \]

Key concepts

Right Circular Cone

A right circular cone is a three-dimensional geometric shape that has a circular base and a single vertex (or apex) opposite the base. **Right** in the context of geometry, indicates a 90-degree angle. In a right circular cone, the axis running from the center of the base to the apex is perpendicular to the base.
This unique property makes calculating its geometrical characteristics, such as volume, more straightforward.
It’s essential to visualize the cone to understand its dimensions:

• **Base radius (r)**: The distance from the center of the base to any point on the circumference.
• **Height (h)**: The perpendicular distance from the apex to the center of the base.

These two primary dimensions are integral when using volume formulas and understanding how different aspects of the cone relate to each other.

Volume Formula

The volume \(V\) of a right circular cone is calculated using the formula: \[ V = \frac{1}{3} \, \pi \, r^2 \, h \] This formula effectively determines how much space is inside the cone. Breaking down the components, we have:

• \(\pi\): A mathematical constant approximately equal to 3.14159.
• \(r^2\): The area of the base, which is a circle.
• \(h\): The height of the cone, which gives the 3D aspect.

When you multiply the base area by the height and then take one-third of that product, you obtain the volume of a cone.
This formula is derived from the basic understanding that the volume of a cone is one-third the volume of a cylinder with the same base and height. It's a crucial formula in understanding the geometry of not only cones but also other related shapes.

Height-Radius Relationship

Understanding the height-radius relationship is key to solving many problems involving cones. In the exercise, the given relationship is that the height (h) is twice the radius (r). We can express this as:\[ h = 2r \] When solving problems, it's beneficial to substitute this relationship back into the volume formula of a cone. For our specific problem, doing so looks like this: \[ V = \frac{1}{3} \, \pi \, r^2 \, (2r) \] Simplifying the equation, we get: \[ V = \frac{1}{3} \, \pi \, 2r^3 = \frac{2}{3} \pi \, r^3 \] This allows us to express the volume solely as a function of the radius.
It's a powerful approach because it simplifies the problem and connects different dimensions of the cone through algebraic substitution. Understanding this relationship can also help visualize how changes in the radius impact the volume, providing a deeper conceptual clarity.