Problem 110 - College Algebra with Modeling and Visualization

Question

The volume \(V\) of a cylinder with a fixed height is directly proportional to the square of its radius \(r\). If a cylinder with a radius of 10 inches has a volume of 200 cubic inches, what is the volume of a cylinder with the same height and a radius of 5 inches?

Step-by-Step Solution

Answer

The volume is 50 cubic inches.

Step 1: Understanding the Relationship

The problem states that the volume \(V\) of a cylinder is directly proportional to the square of its radius \(r\). This can be written as \(V = k \times r^2\), where \(k\) is the proportionality constant.

Step 2: Determine the Proportionality Constant

We know that a cylinder with a radius of 10 inches has a volume of 200 cubic inches. Plug these values into the equation to find \(k\): \(200 = k \times 10^2\). Solving for \(k\), we get \(k = \frac{200}{100} = 2\).

Step 3: Finding the Volume for Radius 5

With \(k = 2\), we can now find the volume of a cylinder with a radius of 5 inches. Use the equation \(V = k \times r^2\): \(V = 2 \times 5^2 = 2 \times 25 = 50\).

Key Concepts

Understanding the Volume of a Cylinder

The volume of a cylinder is an important concept and is calculated using the relationship between the radius, height, and a constant that involves pi (π). In general, the formula for the volume of a cylinder is given by:

• \[ V = \pi \times r^2 \times h \]

This formula states that the volume \( V \) is the product of the area of the circular base and the height \( h \) of the cylinder.
For a fixed height cylinder, if you're only changing the radius \( r \), the volume changes based on the square of the radius. This is why we sometimes say the volume is directly proportional to the square of the radius when the height remains constant.
This direct proportionality simplifies calculations in many practical problems, such as resizing containers in manufacturing, engineering designs, or even kitchen measures.

The Role of Proportionality Constant

In any direct proportion equation, the proportionality constant \( k \) is crucial as it serves as a bridge between variables. For cylinders, when volume \( V \) relates to the square of the radius \( r \), the formula is:

• \[ V = k \times r^2 \]

Here, \( k \) helps us understand how much volume changes with a change in radius for a specific set of conditions (like a fixed height). Knowing \( k \) allows solving for unknowns, like volumes of cylinders with different radii.
Calculating the constant involves using known values. Using our example, we had a volume of 200 cubic inches for a radius of 10 inches, leading to:

• \[ 200 = k \times 10^2 \]

Solving gives \( k = 2 \). This means for every square inch increase in the radius squared, the volume grows twofold under the same conditions.

Understanding Cylinder Radius

The radius \( r \) is simply the distance from the center of the cylinder's base to its edge. In circular objects, the radius plays a vital role in determining size and capacity. Since the area of the base of a cylinder is \( \pi \times r^2 \), changing the radius significantly affects the overall volume.
In situations where the height is constant, like in our exercise, the volume depends purely on changes to the radius. A smaller radius results in a smaller base area, and thus, less volume. Conversely, a larger radius increases both base area and volume.
If a cylinder has a radius of 10 inches with a volume of 200 cubic inches, when the radius is halved to 5 inches, the area and corresponding volume are reduced. By applying the formula \( V = k \times r^2 \) with \( k = 2 \), halving the radius from 10 to 5 leads to a quarter of the starting volume, precisely 50 cubic inches, reflecting how powerful the radius's effect on volume is.