Problem 47 - Algebra 2

Question

GEOMETRY For Exercises 46 and \(47,\) use the following information. The formula for the surface area of a cylinder with radius \(r\) and height \(h\) is \(\pi\) times twice the product of the radius and height plus twice the product of \(\pi\) and the square of the radius. Write an equivalent expression using the Distributive Property.

Step-by-Step Solution

Answer

The equivalent expression is \( 2\pi(rh + r^2) \).

Step 1: Understanding the Surface Area Formula

The given formula for the surface area of a cylinder is \( A = 2\pi rh + 2\pi r^2 \). Here, \( 2\pi \) is a common factor in both terms.

Step 2: Factoring Out the Common Factor

Apply the Distributive Property by factoring out \( 2\pi \) from the expression. The formula becomes \( A = 2\pi(rh + r^2) \).

Step 3: Expression Rewritten

The equivalent expression using the Distributive Property is \( 2\pi(rh + r^2) \), where the common factor \( 2\pi \) is factored out and multiplied by the sum of \( rh \) and \( r^2 \).

Key Concepts

Surface Area

The concept of surface area in geometry refers to the total area of the surface of a three-dimensional object. In the case of a cylinder, the surface area includes not only the sides of the cylinder but also the areas of the two circular bases. Understanding how to calculate this is crucial in various geometric problems.

The surface area of a cylinder, denoted as \( A \), can be found using the formula:

• \( A = 2\pi rh + 2\pi r^2 \)

Where:

• \( r \) is the radius of the circular base.
• \( h \) is the height of the cylinder.
• \( \pi \) (Pi) is a constant approximately equal to 3.14159, representing the ratio of the circumference to the diameter of a circle.

In this formula, \( 2\pi rh \) represents the area of the curved surface (the side) of the cylinder, while \( 2\pi r^2 \) accounts for the combined area of the two circular bases. Calculating the surface area is essential for tasks such as determining the amount of material needed to cover a cylinder or the paint required for its surface.

Distributive Property

The distributive property is a fundamental principle in algebra that shows how multiplication interacts with addition. It allows you to multiply a number by a group of numbers added together in a specific way, simplifying expressions and equations.

In mathematical terms, the distributive property is expressed as:

• \( a(b + c) = ab + ac \)

Applying this property helps to simplify complex expressions, making calculations more manageable.

In the context of the surface area of a cylinder, employing the distributive property involves factoring out any common factors from an expression. For instance, in the expression for the cylinder's surface area \( 2\pi rh + 2\pi r^2 \), \( 2\pi \) is a common factor.
By applying the distributive property, you factor out \( 2\pi \) to rewrite the equation as:

• \( A = 2\pi(rh + r^2) \)

This new expression can sometimes make it easier to understand or solve further problems involving the cylinder's surface area.

Cylinder

A cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface. Cylinders are common in everyday life, found in objects such as cans, pipes, and tubes. Understanding cylinders is essential in various fields including mathematics, physics, and engineering.

The defining features of a cylinder include:

• Two circular bases of equal size, separated by a certain distance referred to as the height \( h \).
• A radius \( r \) which is the distance from the center to the edge of the circular base.
• A lateral or curved surface that connects the two bases.

One key aspect of cylinders is calculating metrics such as volume and surface area. The cylinder's volume involves filling the space within, while surface area pertains to the covering of the exterior. Mastering these measurements is not just theoretical; they are practical for designing and manufacturing cylindrical objects effectively.