Problem 454 - Precalculus

Question

Find the dimensions of the right circular cylinder described. The radius is \(\frac{1}{3}\) meter greater than the height. The volume is \(\frac{98}{9} \pi\) cubic meters.

Step-by-Step Solution

Answer

Height is 2 m, radius is \(\frac{7}{3}\) m.

Step 1: Define the Variables

First, let's define the variables we will use. Let the height of the cylinder be denoted as \( h \) in meters. Since the radius is \( \frac{1}{3} \) meter greater than the height, the radius \( r \) can be expressed as \( r = h + \frac{1}{3} \).

Step 2: Write the Volume Formula for a Cylinder

The formula for the volume \( V \) of a right circular cylinder is given by \( V = \pi r^2 h \). We are given that the volume is \( \frac{98}{9} \pi \).

Step 3: Substitute the Values into the Volume Equation

Substitute the expression for \( r \) in terms of \( h \) and the given volume into the volume formula: \[ \pi (h + \frac{1}{3})^2 h = \frac{98}{9} \pi \].

Step 4: Simplify the Equation

Cancel \( \pi \) from both sides of the equation, and expand the expression:\[ (h + \frac{1}{3})^2 h = \frac{98}{9} \].Expand \( (h + \frac{1}{3})^2 \):\[ (h + \frac{1}{3})^2 = h^2 + \frac{2}{3}h + \frac{1}{9} \].

Step 5: Formulate the Polynomial Equation

Now, multiply the expanded form by \( h \):\[ h(h^2 + \frac{2}{3}h + \frac{1}{9}) = \frac{98}{9} \].Simplify:\[ h^3 + \frac{2}{3}h^2 + \frac{1}{9}h = \frac{98}{9} \].

Step 6: Clear the Fractions

Multiply through the equation by 9 to clear the fractions:\[ 9h^3 + 6h^2 + h = 98 \].

Step 7: Solve the Cubic Equation

Rearrange the equation into standard form:\[ 9h^3 + 6h^2 + h - 98 = 0 \].Use a numerical method or factoring to find the roots. By attempting rational roots, you find that \( h = 2 \) is a solution.

Step 8: Find the Radius Using Height

Having found \( h = 2 \), substitute back to find \( r \):\[ r = h + \frac{1}{3} = 2 + \frac{1}{3} = \frac{7}{3} \].

Key Concepts

Understanding Cubic Equations

A cubic equation is a polynomial equation of degree three, which means it contains at least one term with an exponent of three. In the context of the cylinder exercise, our cubic equation is derived from the expression of the cylinder's volume.
The equation given is:
\[9h^3 + 6h^2 + h - 98 = 0\]
This equation represents the balance of variables involved in finding the cylinder's dimensions, where each term represents a different moment in this balance.
Solving cubic equations like this one might require factoring, using synthetic division, or numerical approaches to find values like the height of the cylinder.
For this problem, we found that the solution is when the height (\(h\)) is equal to 2 meters.

Exploring Polynomial Equations

Polynomial equations are algebraic expressions that sum multiple terms of different powers of a variable, most commonly denoted as \(x\) or in our case \(h\).
Each term can include coefficients and a variable raised to an exponent. Our example here includes a cubic equation with powers 3, 2, and 1 in its terms, making it a rich part of polynomial mathematics.
The equation used for our cylinder problem after clearing fractions is shown as:
\[9h^3 + 6h^2 + h = 98\]
Polynomial equations are crucial in describing real-world phenomena like the volume of geometric shapes. They require correct formulation and transformation to reach a solution.

• Degree: Indicates the highest power of the variable (like cubic for \(h^3\)).
• Coefficients: Numbers multiplying each term (like 9, 6, and 1).
• Solving: Involves strategies like factoring, substitution, or numerical methods.

To solve such polynomial equations, various techniques can be applied depending on their complexity.

Characterizing Right Circular Cylinders

A right circular cylinder is a geometric shape with two parallel circular bases connected by a curved surface, where the axis joining the centers of the two bases is perpendicular to the bases. This perpendicular nature is what makes it a "right" cylinder.
It can be imagined as a perfect roll of paper towel or a can. The critical characteristics of this shape are:

• Height (\(h\)): Distance between the two circular bases.
• Radius (\(r\)): Distance from the center to the edge of the circular base.
• Volume Formula: Given by \(V = \pi r^2 h\), illustrating how both radius and height affect the space within the cylinder.

In solving the exercise, understanding that our cylinder is right and circular helps dictate how we approach writing its volume equation.

Analyzing the Radius and Height Relationship

The relationship between the radius and height is vital in solving for the dimensions of a cylinder efficiently. In this problem, the radius is defined in relation to the height \(h\), specifically\(r = h + \frac{1}{3}\).
This relational expression combines both radius and height into a single variable \(h\), allowing for easy substitution into the volume formula, facilitating streamlined calculations.

• Increases Simplicity: Reduces two unknowns into one, making equations more manageable.
• Relies on Geometry: Knowledge of distances and axis placement contributes to forming relationships.
• Volume Efficiency: Directly influences calculations for geometric volumes like our cylinder.

Recognizing and applying these relationships is essential in eliminating work redundancy and boosting problem-solving effectiveness.