Problem 300 - Precalculus

Question

For the following exercises, write the polynomial function that models the given situation. A cylinder has a radius of \(x+2\) units and a height of 3 units greater. Expres of the cylinder as a polynomial function.

Step-by-Step Solution

Answer

The volume as a polynomial function is \( V = \pi(x^3 + 9x^2 + 24x + 20) \).

Step 1: Identify the formula for the volume of a cylinder

The formula for the volume of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cylinder. We need to substitute the expressions for \( r \) and \( h \) into this formula.

Step 2: Define the expressions for radius and height

According to the problem, the radius \( r \) of the cylinder is \( x+2 \) units. The height \( h \) is said to be 3 units greater than the radius, which means it is \( x+2+3 = x+5 \) units.

Step 3: Set up the equation for the volume

Substitute the expressions for \( r \) and \( h \) into the volume formula: \[ V = \pi (x+2)^2 (x+5) \].

Step 4: Expand the radius squared term

Calculate \( (x+2)^2 \) by expanding the binomial: \( (x+2)^2 = x^2 + 4x + 4 \).

Step 5: Multiply to find the volume polynomial

Substitute the expanded form from Step 4 into the volume equation: \[ V = \pi (x^2 + 4x + 4)(x+5) \]. Expand this product:\\( V = \pi [(x^2 + 4x + 4)x + (x^2 + 4x + 4)5] \).\Continue expanding: \[ V = \pi(x^3 + 4x^2 + 4x + 5x^2 + 20x + 20) \]. Simplify to obtain: \[ V = \pi (x^3 + 9x^2 + 24x + 20) \].

Key Concepts

Volume of a Cylinder

The volume of a cylinder is an important concept in both geometry and algebra, providing a bridge between the shapes we see around us and their mathematical representations. A cylinder is essentially a three-dimensional shape with two parallel circular bases and a curved surface that connects these bases. To find the volume, which is the amount of space enclosed within the cylinder, we use a formula that incorporates both the area of the circular base and the height of the cylinder.

The formula for calculating the volume of a cylinder is:

• \( V = \pi r^2 h \)

where:

• \( \pi \) is a mathematical constant (approximated as 3.14159)
• \( r \) represents the radius of the cylinder's base
• \( h \) denotes the height of the cylinder

This formula shows the fine interplay between geometry—using properties of circles and linear dimensions—and algebra, as it involves substituting values and variables to find an expression for an unknown quantity. Understanding how each term affects the volume can help students visualize how changes in the cylinder's dimensions alter the whole structure.

Expanding Binomials

Expanding binomials is a fundamental algebraic skill that involves writing an expression in an expanded form by multiplying terms together. In the given problem, expanding the binomial expression is a critical step toward calculating the volume of the cylinder as a polynomial function.

The specific task involves expanding the squared term of the radius, \((x+2)^2\). This is done using a method known as binomial expansion, which can be performed as follows:

• Recognize that \((x+2)^2\) is shorthand for \((x+2) \times (x+2)\)
• Apply the distributive property: \(x(x+2) + 2(x+2)\)
• Calculate each term: \(x^2 + 2x + 2x + 4\)
• Simplify the expression: \(x^2 + 4x + 4\)

This expanded form is then used to further develop the polynomial that represents the cylinder's volume. Mastering the technique of expanding binomials is crucial for solving polynomial equations and is widely applicable in various mathematical contexts.

Mathematical Modeling

Mathematical modeling is a powerful tool that allows us to represent real-world situations using mathematical constructs. In the exercise at hand, we are tasked with expressing the volume of a cylinder as a polynomial function by combining different mathematical concepts.

This scenario involves taking a geometric problem and translating it into an algebraic expression—a process typical of mathematical modeling, which often requires:

• Identifying pertinent variables and constants (e.g., radius and height)
• Using existing formulas (e.g., volume formula for a cylinder)
• Substituting expressions involving these variables into those formulas
• Performing algebraic manipulations (like expanding and simplifying)

The goal is to create an understandable and usable mathematical representation, which in this case is a polynomial that describes the volume given the expressed dimensions of the cylinder. Mathematical modeling not only helps solve specific tasks but also develops a deeper understanding of how mathematical theories apply in practical settings.

Geometry in Algebra

Geometry in algebraic expressions reveals the underlying relationships between shapes and equations, enhancing our ability to describe and analyze spatial problems mathematically. In the problem of finding the volume of a cylinder, we see a seamless transition from geometric features to algebraic terms.

The radius and height of a cylinder are geometric measurements, but when expressed with algebraic variables (like \(x\)), we begin to see geometry behaving through algebra's lens. This connection enables the manipulation of dimensions as if they were part of a simple numerical calculation, thus creating a polynomial expression for volume.

In this context:

• The algebraic expression \((x+2)\) represents a geometric concept (radius)
• The polynomial derived \((\pi(x^3 + 9x^2 + 24x + 20))\) encapsulates the volume

Exploring geometry in algebra demystifies many spatial problems and strengthens the understanding of how dimensions translate into mathematical language, preparing students to bridge concepts for solving real-world applications.