Problem 450 - Precalculus

Question

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

The height is 1 inch and the radius is 4 inches.

Step 1: Understand the problem

The problem states that we have a right circular cylinder where the radius is 3 inches more than the height. We are given the volume of the cylinder as \(16\pi\) cubic meters and need to find the dimensions of the cylinder (both height and radius).

Step 2: Define the variables

Let \(h\) represent the height of the cylinder in inches. Since the radius is 3 inches more than the height, the radius \(r\) is \(h + 3\) inches.

Step 3: Write the formula for the volume of the cylinder

The formula for the volume \(V\) of a cylinder is given by \(V = \pi r^2 h\). Here, \(V = 16\pi\), \(r = h + 3\). Substitute these into the formula to get \(16\pi = \pi (h + 3)^2 h\).

Step 4: Remove \(\pi\) by dividing both sides

Divide both sides of the equation \(16\pi = \pi (h + 3)^2 h\) by \(\pi\) to simplify the equation: \(16 = (h + 3)^2 h\).

Step 5: Expand and solve the equation

Expand \((h + 3)^2\) to get \(h^2 + 6h + 9\). Thus, the equation becomes \(16 = (h^2 + 6h + 9)h\). Simplify to \(16 = h^3 + 6h^2 + 9h\).

Step 6: Rearrange and solve for \(h\)

Rearrange the equation to form a cubic equation: \(h^3 + 6h^2 + 9h - 16 = 0\). Solve this equation to find acceptable values for \(h\).

Step 7: Solving the cubic equation

Testing small integer values, we find that when \(h = 1\), the cubic equation becomes \(1^3 + 6 \cdot 1^2 + 9 \cdot 1 - 16 = 0\), which holds true. So, \(h = 1\) inch.

Step 8: Find the radius

Since \(h = 1\) inch, the radius \(r = h + 3 = 1 + 3 = 4\) inches.

Key Concepts

Right Circular Cylinder

A right circular cylinder is a three-dimensional shape defined by having two parallel circular bases connected by a curved surface. These bases are equal in size, making the shape perfectly symmetrical. What makes it 'right' is that the axis, or the imaginary line connecting the centers of the two bases, is perpendicular to the bases themselves. This is what most people visualize when they think of a can or a drum.

• Each base has a radius, denoted as \(r\), which is the distance from the center to the edge of the circle.
• The height \(h\) is the straight distance between the bases.

Understanding this shape is crucial when visualizing problems related to volume, surface area, or any physical property that depends on its geometric dimensions. Knowing the cylinder's volume helps us understand the capacity it can hold, a fundamental aspect in both mathematical problems and real-life applications.

Radius and Height Relationship

In some problems involving cylindrical shapes, the radius and height might be related to each other algebraically. Here, the radius \(r\) is stated as being 3 inches more than the height \(h\). This can be written mathematically as:

• \(r = h + 3\)

This relationship simplifies calculations and helps set up equations to solve for unknown dimensions. It shows how one variable relates to another, making the problem easier by reducing it to one variable. This concept is essential in understanding how changes in dimensions affect the overall object."
If you know the relationship between the radius and height, you can more easily solve problems involving complex measurements or different units of measure. In everyday scenarios, such as designing a cylindrical container with specific constraints, knowing how height and radius relate can simplify the design process.

Cubic Equation

Cubic equations can appear in geometry when dealing with polynomial expressions of degree three. They often arise when solving for dimensions in problems with volume or surface area constraints, as was the case here.
The cubic equation derived from this problem is:

• \(h^3 + 6h^2 + 9h - 16 = 0\)

This equation came from substituting the known values and relationships into the formula for the volume of a cylinder \(V = \pi r^2 h\), and then simplifying.
Solving cubic equations can be more challenging than linear or quadratic equations, but similar techniques apply:

• Look for possible integer solutions by "guess and check" or testing small values, often beginning with zeros of related quadratic factors.
• Use synthetic division once a root is found to factor the cubic equation further.

These solutions help identify feasible dimensions for the cylinder, allowing us to determine height and radius from real-world measurements.