Problem 78 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Common Factor

The given formula for the surface area of the cone is \( SA = \pi r^2 + \pi r s \). Notice that both terms share the common factor \( \pi r \).

Step 2: Factor Out the Common Term

To express the formula in factored form, factor out the common term \( \pi r \) from both terms: \( SA = \pi r (r + s) \).

Step 3: Verify the Factorization

To verify, distribute the factored term \( \pi r \) back into the parentheses: \( \pi r (r + s) = \pi r \cdot r + \pi r \cdot s = \pi r^2 + \pi r s \). This confirms the factorization is correct.

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that makes expressions simpler and more manageable. It involves identifying common elements within terms to simplify an equation or an expression. For example, in the surface area formula of a cone, \( SA = \pi r^2 + \pi r s \), both terms share a common factor \( \pi r \). This means that each term can be divided by \( \pi r \) without changing the equality of the expression.

By factoring out the common term, the expression becomes more compact: \( SA = \pi r (r + s) \). Factoring not only simplifies expressions but also helps in solving equations and finding solutions. It is a critical skill in algebra that aids in understanding the relationships between different terms in an expression.

Surface Area

The surface area of a cone is a measure of how much exposed area the cone has. Surface area is important in real-life applications, like figuring out how much material is needed to make a cone-shaped object. The formula used to calculate the surface area of a cone is \( S A = \pi r^2 + \pi r s \), where \( r \) is the radius of the base, and \( s \) is the slant height of the cone.

The first part of the formula, \( \pi r^2 \), calculates the area of the circular base. The second part, \( \pi r s \), finds the area of the cone's lateral surface (the "sides" of the cone). When combined, these terms give you the total surface area. Understanding and using this formula is crucial in geometry when dealing with cones or similar shapes.

Geometry

Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. It allows you to understand and calculate measurements such as areas and volumes of different shapes. In the context of a cone, you encounter geometric concepts like the base, radius, height, and slant height.

Knowing the geometric properties of a cone helps when applying various formulas to compute dimensions such as volume or surface area. Geometry provides a visual understanding, making it easier to see how different formulas connect to the physical shapes they represent. This geometric understanding is essential for solving real-world problems that involve spatial reasoning.

Formulas

Formulas in mathematics are concise ways to express complex ideas or calculations. They serve as recipes that tell you how to perform a particular calculation. In the case of the cone's surface area, the formula \( SA = \pi r^2 + \pi r s \) gives the surface area by combining the base and lateral surface area components into one expression.

Using formulas ensures consistency and accuracy across different calculations. They are reusable and allow for quick computations once all variables are known. Understanding how to manipulate and factor these expressions, as demonstrated here, can help simplify calculations and reveal underlying patterns in mathematical problems.