Problem 77 - College Algebra with Corequisite Support

Question

For the following exercises, find the dimensions of the right circular cylinder described. The height is one less than one half the radius. The volume is \(72 \pi\) cubic meters.

Step-by-Step Solution

Answer

The radius is 6 meters and the height is 2 meters.

Step 1: Identifying Variables and Relationships

Let the radius of the cylinder be \( r \) meters. According to the problem, the height \( h \) of the cylinder is one less than one half of the radius, so we can express this as \( h = \frac{r}{2} - 1 \).

Step 2: Formulating the Volume Equation

The volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \). We know from the problem that the volume is \( 72 \pi \). Substituting the expression for \( h \) into the volume formula gives: \( 72 \pi = \pi r^2 \left( \frac{r}{2} - 1 \right) \).

Step 3: Simplifying the Volume Equation

Divide both sides of the equation \( 72 \pi = \pi r^2 \left( \frac{r}{2} - 1 \right) \) by \( \pi \) to obtain: \( 72 = r^2 \left( \frac{r}{2} - 1 \right) \).

Step 4: Solving for the Radius \( r \)

Expand the equation: \( 72 = \frac{r^3}{2} - r^2 \). Multiply the entire equation by 2 to eliminate the fraction: \( 144 = r^3 - 2r^2 \). Rearrange it to: \( r^3 - 2r^2 - 144 = 0 \).

Step 5: Factoring the Cubic Equation

Use trial and error or synthetic division to find that \( r = 6 \) is a root. Factor the equation as \( (r - 6)(r^2 + 4r + 24) = 0 \). The quadratic \( r^2 + 4r + 24 \) does not have real roots (no real solutions can be found using the quadratic formula), so the only real root is \( r = 6 \).

Step 6: Calculating the Height \( h \)

Since we have \( r = 6 \), substitute this back into the formula for height: \( h = \frac{6}{2} - 1 = 3 - 1 = 2 \). Therefore, the height of the cylinder is \( 2 \) meters.

Key Concepts

Cylindrical Geometry

Cylindrical geometry involves studying the shape and properties of a cylinder, which is a three-dimensional geometric figure with parallel circular bases connected by a curved surface. In problems involving cylinders, you need to understand certain basic traits:

• The two circular bases are identical and parallel.
• The height (or length) of the cylinder is the perpendicular distance between the bases.
• The radius is the distance from the center to the edge of the base.

To solve problems, you'll often work with a right circular cylinder, meaning the sides are perpendicular to the base, forming right angles. This simplifies calculations and is the most common type encountered in algebra and geometry. Exploring cylindrical geometry involves measurements like radius, height, and volume. Remember, the formula for the volume of a cylinder is crucial: \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.

Algebraic Equations

Algebraic equations are mathematical statements that use symbols and numbers to express the relationships between different quantities. In our cylinder problem, we use algebraic equations to find unknown dimensions. Here’s a breakdown:

• An algebraic equation consists of two expressions separated by an equal sign.
• In the cylinder volume problem, setting the formula for volume equal to the given volume creates an equation to solve.
• The expression \( h = \frac{r}{2} - 1 \) translates the problem's verbal relationship into an algebraic form.

Solving equations often requires manipulating them by balancing both sides through operations like addition, subtraction, and most frequently here, multiplication or division. Simplifying the volume equation involves altering its structure for easier resolution, ensuring each term's logical consistency is maintained.

Problem Solving in Mathematics

Problem solving in mathematics involves a series of logical steps that lead to the solution of a given problem. For the cylinder problem, here's how this approach aids understanding and solution:

• First, interpret the problem by identifying knowns and unknowns, as demonstrated with variables \( r \) and \( h \).
• Next, use formulas and relationships pertinent to the geometry of the cylinder to structure equations that describe the scenario.
• Simplification and manipulation of these equations help isolate and solve for unknowns.
• Finally, validate the solution by checking if it satisfies the original conditions of the problem.

Approaching mathematical problems systematically helps build confidence and efficiency. By carefully verifying each step, from understanding through to validation, problem solving becomes not just a task but a methodical, rewarding process.