Problem 76 - College Algebra

Question

For the following exercises, find the dimensions of the right circular cylinder described. The radius is 3 inches more than the height. The volume is 16\(\pi\) cubic meters.

Step-by-Step Solution

Answer

Height = 1 meter, Radius = 4 meters

Step 1: Understand the Problem

We need to find the dimensions of a right circular cylinder, given that the radius is 3 inches more than the height, and the volume is \(16\pi\) cubic meters. We will use the volume formula for a cylinder, \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height.

Step 2: Define Variables

Let the height of the cylinder be \(h\) meters. Then the radius can be expressed as \(r = h + 3\) meters according to the problem's condition.

Step 3: Set Up the Volume Equation

Substitute the expressions for the radius and height into the volume formula:\[V = \pi (h + 3)^2 h = 16\pi\]

Step 4: Simplify and Solve for Height

Cancel \(\pi\) from both sides and simplify the equation:\[(h + 3)^2 h = 16\]Expanding the equation gives:\[(h^2 + 6h + 9)h = 16\]Thus:\[h^3 + 6h^2 + 9h = 16\]Simplify the equation to find \(h\):\[h^3 + 6h^2 + 9h - 16 = 0\]Testing for rational roots or using numeric methods gives \(h = 1\).

Step 5: Find the Radius

With \(h = 1\) meters, substitute back into the equation for the radius:\[r = h + 3 = 1 + 3 = 4 \text{ meters}\]

Step 6: Verify the Solution

Substitute \(r = 4\) meters and \(h = 1\) meter back into the volume formula:\[V = \pi r^2 h = \pi (4^2)(1) = 16\pi \text{ cubic meters}\]This confirms that our solution satisfies the volume condition.

Key Concepts

Cylinder Dimensions Explained

A cylinder is characterized by two main dimensions: the radius and the height. These dimensions are essential in defining the shape and size of the cylinder. The radius is the distance from the center of the base to the edge, while the height is the distance between the two circular bases. Understanding these dimensions is crucial in applying formulas to calculate volume or surface area.

In mathematical problems, it is common to encounter situations where one dimension relies on another, much like in our problem where the radius is 3 inches more than the height. Using relationships between dimensions can simplify complex problems, making it easier to set up equations and solve them. When approaching a problem involving cylinder dimensions, it is helpful to:

• Identify given values and relationships between dimensions.
• Express unknown dimensions in terms of a variable, like using "h" for height.
• Apply these expressions in relevant formulas.

Understanding a Right Circular Cylinder

A right circular cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. The axis of the cylinder is perpendicular to the center of these bases, which distinguishes it from cylinders that might have tilted or elliptical bases.

Right circular cylinders are often used in practical applications due to their simple and symmetrical structure, making calculations straightforward. This type of cylinder:

• Has circular bases, which means both bases are congruent circles.
• Is called 'right' because the axis is perpendicular to the bases.
• When picturing a right circular cylinder, imagine a soup can standing upright.

The geometric simplicity means calculations such as volume or surface area can be achieved without dealing with additional complexities that might arise in other types of three-dimensional objects.

Deriving and Using the Volume Formula

The volume of a cylinder is a measure of how much space it occupies. For a right circular cylinder, the volume formula is given by: \( V = \pi r^2 h \). In this formula:

• \( V \) denotes the volume.
• \( \pi \) is a constant approximately equal to 3.14159...
• \( r \) represents the radius of the circular base.
• \( h \) is the height of the cylinder.

The formula \( V = \pi r^2 h \) showcases the relationship between the dimensions of the cylinder and its volume. It combines the area of a circle \( \pi r^2 \) with height \( h \), reflecting the layering of multiple circles (or disks) along the height of the cylinder.

In our problem, after defining relationships between the dimensions, we substituted these expressions into the volume formula, solved for the unknown dimension, the height \( h \), and confirmed it using the known volume. Such problems often involve solving equations resulting from substituting known or given conditions into geometric formulas.