Functions - Volume As A Function Of Radius

🔑 Key Concepts

• The volume of a cone is calculated using \(\displaystyle V=\frac{1}{3}\pi r^2h\).
• When the height remains constant, cone volume can be viewed as a function of the radius.
• The radius is the input, while the volume is the output.
• Because the radius is squared, multiplying the radius by \(k\) multiplies the volume by \(k^2\).
• The factor \(\displaystyle \frac{1}{3}\) changes the size of every output but does not change the quadratic relationship.
• The graph of volume as a function of radius is curved, so the relationship is nonlinear.

✏️ Worked Examples

What Changes and What Stays Fixed?

Both cones have the same height:

\(h=6\text{ cm}\)

The radius changes from \(1\) centimeter to \(2\) centimeters. Therefore, the radius is the independent variable, and volume is the dependent variable.

Input: \(r\)
Output: \(V(r)\)
Fixed parameter: \(h=6\)

Writing the Cone Volume Function

Begin with the volume formula for a cone:

\(\displaystyle V=\frac{1}{3}\pi r^2h\)

Since the height is fixed at \(6\) centimeters, substitute \(6\) for \(h\):

\(\displaystyle V(r)=\frac{1}{3}\pi r^2(6)\)

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

The notation \(V(r)\) shows that the volume depends on the radius.

Volume function:
\(\boxed{V(r)=2\pi r^2}\)

Doubling the Radius

Find the volume when \(r=1\) centimeter:

\(V(1)=2\pi(1)^2\)

\(V(1)=2\pi\text{ cm}^3\)

Now double the radius to \(2\) centimeters:

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

\(V(2)=8\pi\text{ cm}^3\)

\(\displaystyle \frac{8\pi}{2\pi}=4\)

Radius scale factor: \(2\)
Volume scale factor: \(2^2=4\)

Why Does the Volume Become Four Times as Large?

A cone’s volume can also be understood as one-third of the area of its base multiplied by its height:

\(\displaystyle V=\frac{1}{3}Bh\)

In this example, the height is held constant. Therefore, the changing part of the formula is the circular base area:

\(B=\pi r^2\)

When the radius doubles, the two-dimensional circular base becomes:

\(\pi(2r)^2=4\pi r^2\)

The base area becomes four times as large. Since the cone’s height and the factor \(\displaystyle \frac{1}{3}\) remain constant, the cone’s volume also becomes four times as large.

General Scale-Factor Rule

Suppose the radius changes from \(r\) to \(kr\), while the height stays fixed.

\(\displaystyle V(r)=\frac{1}{3}\pi r^2h\)

\(\displaystyle V(kr)=\frac{1}{3}\pi(kr)^2h\)

\(\displaystyle V(kr)=\frac{1}{3}\pi k^2r^2h\)

\(\displaystyle V(kr)=k^2\left(\frac{1}{3}\pi r^2h\right)\)

\(\boxed{V(kr)=k^2V(r)}\)

Multiplying the radius by \(k\) multiplies the cone’s volume by \(k^2\), as long as the height remains fixed.

Interpreting the Graph

In the video, cones have a fixed height of \(3\) inches. Substitute \(3\) into the cone volume formula:

\(\displaystyle V(r)=\frac{1}{3}\pi r^2(3)\)

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

This is a quadratic function of the radius. Its graph curves upward rather than forming a straight line.

Statement	True or False?	Reason
If the radius doubles, the volume is multiplied by \(4\).	True	\(2^2=4\)
The relationship is linear.	False	The graph is curved, and the radius is squared.
The relationship is nonlinear.	True	\(V(r)=\pi r^2\) is quadratic.
If the radius doubles, the volume doubles.	False	The volume becomes four times as large.
When \(r=2.5\), the volume is about \(20\text{ in}^3\).	True	\(\pi(2.5)^2\approx19.6\)

🧠 Math Vocabulary

• Function: A relationship that assigns exactly one output to each input.
• Input: The independent value entered into a function. In this lesson, the radius is the input.
• Output: The value produced by a function. In this lesson, the volume is the output.
• Independent variable: The variable selected or changed. The radius is the independent variable.
• Dependent variable: A variable whose value depends on another quantity. Cone volume depends on radius.
• Constant parameter: A quantity held fixed while another quantity varies. In this lesson, the height is constant.
• Quadratic function: A function in which the independent variable is raised to the second power.
• Nonlinear relationship: A relationship whose graph is not a straight line and whose rate of change is not constant.
• Scale factor: A multiplier that describes how much a measurement is enlarged or reduced.
• Radius: The distance from the center of a circular base to its outer edge.
• Height: The perpendicular distance from the circular base of a cone to its vertex.
• Base area: The area of the circular base, found using \(B=\pi r^2\).
• Volume: The amount of three-dimensional space inside a solid.
• Cubic units: Units used to measure volume, such as cubic inches or \(\text{in}^3\).

💡 Main Idea

When a cone’s height remains fixed, its volume can be modeled as a function of its radius. The radius is the input, and volume is the output. Because the formula contains \(r^2\), multiplying the radius by a scale factor of \(k\) multiplies the volume by \(k^2\). The constant factor \(\displaystyle \frac{1}{3}\) makes a cone’s volume smaller than the volume of a corresponding cylinder, but it does not change the quadratic relationship between radius and volume.

📚 What You Should Already Know

Students should know how to find the area of a circle using \(A=\pi r^2\), calculate the volume of a cone, evaluate squares, substitute values into formulas, interpret input-output relationships, and distinguish between linear and nonlinear graphs.

🚀 What Comes Next

Students can compare the radius-volume functions of cones and cylinders, graph quadratic volume relationships, examine how changing the height affects the function rule, and solve inverse problems in which the volume is known and the missing radius must be determined.

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