Problem 46 - 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 this as an algebraic expression.

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to express the surface area of a cylinder using an algebraic expression. The given information mentions the surface area formula involves the radius \(r\) and height \(h\) of the cylinder.

Step 2: Identify Surface Area Components

For a cylinder, the surface area includes the area of the top and bottom circles and the side (which is a rectangle wrapped around). The formula given is based on these components.

Step 3: Calculate Area of Top and Bottom

Each circular end of the cylinder has an area of \(\pi r^2\). Since there are two circles (top and bottom), their combined area is \(2\pi r^2\).

Step 4: Calculate Lateral Surface Area

The lateral surface area is the area of the rectangle that wraps around the cylinder. The rectangle's height is \(h\), and its width is the circumference of the base \(2\pi r\). Therefore, the lateral surface area is \(2\pi rh\).

Step 5: Combine Components for Total Surface Area

Add the area of the top and bottom to the lateral surface area to get the total surface area. Thus, the algebraic expression is \(2\pi r^2 + 2\pi rh\).

Key Concepts

Geometry Problem-Solving

Solving geometry problems involves understanding the basic components of the shapes you're dealing with. For a cylinder, you'll want to focus on both its circular bases and the rectangle that connects them. In this exercise, we're tasked with finding the surface area—a measure of how much space is covered by the outside of the cylinder. By dissecting the problem into manageable parts (the circles and the rectangle), we can calculate each area separately and combine them.

• Break down the figure into basic shapes (circles and rectangle).
• Determine formulas for the areas of these shapes.
• Combine these areas to find the total surface area.

This problem-solving strategy is invaluable in geometry as it simplifies what can initially seem complex.

Algebraic Expression

An algebraic expression in this context translates a verbal formula into a mathematical symbol-based one. When asked to write the surface area of the cylinder as an algebraic expression, we interpret phrases into mathematical operations using variables and constants.
This particular formula involves:

• Multiplication, which is shown as products of variables and constants (like \(2\pi r^2\)).
• Addition, which combines different components of the formula.

By converting spoken descriptions into an algebraic form, you gain a clearer, more concise way to calculate values. This is crucial in math, providing consistent and quick ways to handle calculations.

Cylinder Surface Area Formula

The surface area formula for a cylinder combines the areas of its two main parts. A cylinder consists of two circles (the top and bottom) and a rectangle wrapped around them.

• The area of the circles: \(2\pi r^2\).
• The area of the side or lateral surface (the rectangle): \(2\pi rh\).

The formula is expressed as \(2\pi r^2 + 2\pi rh\). By understanding each component and its contribution to the total surface area, students can accurately use this formula to solve various geometry problems involving cylinders.

Radius and Height in Geometry

Understanding the radius and height is crucial in geometry, especially when dealing with cylinders. The radius \(r\) is the distance from the center of the circle to its edge, effectively determining the size of the cylinder's bases.
The height \(h\) represents the distance between the two circular bases. Here's why they matter:

• The radius directly influences the size of the base and, by extension, the overall width of the cylinder.
• The height affects the size of the cylinder's side or lateral surface.

When you understand these dimensions, applying them in formulas becomes straightforward, turning complex geometric figures into solvable problems.