Problem 77 - Elementary Algebra

Question

The surface area of a cylinder is given by the formula \(S A=2 \pi r 2+2 \pi r h,\) where \(r\) represents the radius of the base and \(h\) is the height of the cylinder. Express this formula in factored form.

Step-by-Step Solution

Answer

The factored form is \(SA = 2\pi r (r + h)\).

Step 1: Identify common factors

Examine the formula for the surface area of the cylinder: \[ SA = 2 \pi r^2 + 2 \pi rh \]The common factors in both terms are \(2 \pi r\). We will factor these out in the next step.

Step 2: Factor the expression

After identifying the common factor \(2 \pi r\), factor it out from the given expression:\[ SA = 2 \pi r (r + h) \]This shows the original expression in its factored form.

Key Concepts

Surface Area of a Cylinder

Cylinders are fascinating shapes often encountered in geometry and real-world objects like cans or pipes. To grasp their surface area, we must understand two crucial components: the curved surface and the top and bottom circles, known as the bases. The formula for the surface area of a cylinder combines these aspects. It is given by \( SA = 2 \pi r^2 + 2 \pi rh \).
This equation consists of two parts:

• \(2 \pi r^2\): This represents the area of the two circular bases. The factor \(\pi r^2\) is the area of one circle, and since we have two, it’s multiplied by 2.
• \(2 \pi rh\): This is the area of the curved surface around the cylinder. Imagine unwrapping it into a rectangle where the width is the circumference of the base, \(2 \pi r\), and the height is \( h\).

Understanding these components helps you visualize why each part exists in the formula, making it easier to factor and re-arrange if needed. A solid grasp of how to calculate and manipulate this formula is fundamental in various applications, from basic geometry to complex engineering.

Common Factors

When dealing with algebraic expressions, identifying common factors is a vital skill. A common factor is a term that appears in each part of an expression or equation, allowing us to simplify the expression by factoring them out.
For the surface area formula of a cylinder, \(SA = 2 \pi r^2 + 2 \pi rh \),it's clear that both terms share a common factor: \(2 \pi r\). Recognizing this can simplify working with the formula significantly.
Separating out these common factors involves:

• Identifying terms that appear in each component of the expression, here \(2 \pi r\).
• Re-writing the expression by placing the factor outside a parenthesis and leaving the remaining terms inside.

Thus, the factorization of the expression into \(SA = 2 \pi r (r + h) \) illustrates how common factors streamline calculations and make expressions more manageable. This process is especially useful in solving equations and simplifying expressions.

Algebraic Formulas

Algebraic formulas like those used for finding surface areas help translate complex geometric ideas into easy-to-manage numbers and symbols. These formulas form the backbone of many calculations in mathematics and science.
Here, we encounter the formula for the surface area of a cylinder:\(SA = 2 \pi r^2 + 2 \pi rh \),which can be factored to \(SA = 2 \pi r (r + h) \).The beauty of algebraic formulas is their consistency and applicability across different problems. They allow one to predict and understand relationships between different elements, such as the ratio between a cylinder’s radius and its height, and how these factors impact the overall surface area.
By manipulating these formulas, such as factoring them to find common solutions or simplifying computations, you gain deeper insights into how different mathematical elements interact. Mastery of these formulas allows for efficient problem-solving and is indispensable in advanced mathematics and practical applications alike.