Volume Of Cones As A Function Of Height

🔑 Key Concepts

• The volume of a cylinder is calculated using \(V=\pi r^2h\).
• When the height remains constant, the volume can be viewed as a function of the radius.
• The radius is the input, and the cylinder’s volume is the output.
• Because the radius is squared, changing the radius by a scale factor of \(k\) changes the volume by a scale factor of \(k^2\).
• The relationship between radius and volume is nonlinear because the radius is raised to the second power.

✏️ Worked Examples

Radius as the Input

The cylinders have the same height:

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

Only the radius changes. Therefore, the radius is the independent variable, or input, while the volume is the dependent variable, or output.

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

Writing the Volume Function

Begin with the cylinder volume formula:

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

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

\(V(r)=\pi r^2(6)\)

\(V(r)=6\pi r^2\)

The notation \(V(r)\) means that the volume depends on the value chosen for the radius.

Volume function:
\(\boxed{V(r)=6\pi r^2}\)

Doubling the Radius

First, find the volume when \(r=2\) centimeters:

\(V(2)=6\pi(2)^2\)

\(V(2)=24\pi\text{ cm}^3\)

Now double the radius from \(2\) centimeters to \(4\) centimeters:

\(V(4)=6\pi(4)^2\)

\(V(4)=96\pi\text{ cm}^3\)

\(\displaystyle \frac{96\pi}{24\pi}=4\)

The radius doubled:
\(2^2=4\)
The volume became four times as large.

Function Table for \(V(r)=8\pi r^2\)

In the video, the cylinder’s height is fixed at \(8\) inches, so the volume function is:

\(V(r)=\pi r^2(8)=8\pi r^2\)

Radius \(r\)	\(r^2\)	Volume \(V(r)=8\pi r^2\)	Volume Scale Factor
\(1\)	\(1\)	\(8\pi\)	\(1\)
\(2\)	\(4\)	\(32\pi\)	\(4\)
\(3\)	\(9\)	\(72\pi\)	\(9\)
\(4\)	\(16\)	\(128\pi\)	\(16\)
\(5\)	\(25\)	\(200\pi\)	\(25\)
\(6\)	\(36\)	\(288\pi\)	\(36\)

The output values do not increase by a constant difference. Instead, they grow according to the square of the radius.

General Scale-Factor Rule

Suppose a cylinder’s radius changes from \(r\) to \(kr\), while its height remains fixed.

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

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

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

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

Therefore, multiplying the radius by \(k\) multiplies the volume by \(k^2\). For example, tripling the radius makes the volume \(3^2=9\) times as large, while quadrupling the radius makes the volume \(4^2=16\) times as large.

🧠 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 cylinder’s volume is the output.
• Independent variable: The variable whose value is selected or changed. The radius is the independent variable.
• Dependent variable: A variable whose value depends on another quantity. The volume depends on the radius.
• Constant parameter: A quantity held fixed while another variable changes. In this relationship, the cylinder’s height is a constant parameter.
• Scale factor: A multiplier describing how much a measurement is enlarged or reduced.
• Nonlinear function: A function whose graph is not a straight line and whose outputs do not change at a constant rate.
• Radius: The distance from the center of a circular base to its outer edge.
• Height: The perpendicular distance between the two circular bases of a cylinder.
• Volume: The amount of three-dimensional space contained inside a solid.
• Cubic units: Units used to measure volume, such as cubic inches or \(\text{in}^3\).

💡 Main Idea

When a cylinder’s height remains fixed, its volume can be modeled as a function of its radius. The formula \(V(r)=\pi hr^2\) shows that the radius is squared. Therefore, the volume does not increase in direct proportion to the radius. If the radius is multiplied by a scale factor of \(k\), the volume is multiplied by \(k^2\). This creates a nonlinear, quadratic relationship between the radius and volume.

📚 What You Should Already Know

Students should know how to calculate the area of a circle using \(A=\pi r^2\), evaluate squares, substitute values into formulas, multiply expressions involving \(\pi\), interpret input-output tables, and recognize that a function assigns one output to each input.

🚀 What Comes Next

Students can extend this reasoning by examining the volume of cones as a function of the radius, comparing different nonlinear volume relationships, graphing radius-volume functions, and solving inverse problems in which the volume is known and a missing radius or height must be determined.

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