Problem 97 - Algebra for College Students

Question

Answer the question with an algebraic expression. The perimeter of a rectangle is \(y\) yards and \(f\) feet. What is the perimeter expressed in feet?

Step-by-Step Solution

Answer

The perimeter in feet is \(3y + f\).

Step 1: Understand the Problem

The problem involves converting the perimeter of a rectangle given partly in yards and partly in feet, into a single unit: feet. We know that 1 yard equals 3 feet.

Step 2: Convert Yards to Feet

Since there are 3 feet in a yard, we need to convert the perimeter in yards (\( y \text{ yards} \)) to feet. The conversion is given by: \( 3y \text{ feet} \).

Step 3: Add the Feet

Now, add the converted yards to the already given feet. The expression for the total perimeter in feet becomes \( 3y + f \).

Step 4: Formulate the Expression

The final algebraic expression for the perimeter of the rectangle in feet is: \( 3y + f \).

Key Concepts

Unit Conversion

When dealing with measurements, unit conversion becomes an essential skill. Units like yards and feet are used to measure length, and sometimes we need to convert these units to solve mathematical problems. The conversion between yards and feet is straightforward:

• 1 yard is equivalent to 3 feet.

To convert yards into feet, you multiply the number of yards by 3. This allows you to express both measurements consistently, simplifying calculations and making results easily understandable. Unit conversion helps us apply consistent and accurate reasoning across different measurement units. Always remember to verify your conversion factor and apply it correctly to get the right unit in your final expression.

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around its boundary. It's important to grasp this concept as it's used to measure the outline of various shapes. For a rectangle, the formula for the perimeter is:

• The sum of twice the length and twice the width: \[ P = 2 imes ( ext{Length}) + 2 imes ( ext{Width}) \].

In some problems, you might receive given perimeter measurements in different units, as in yards and feet, forcing you to convert these units for consistency. Knowing how to find and manipulate the perimeter provides a foundation for understanding more complex geometric concepts and solving real-world problems. It connects algebra and geometry by frequently appearing in problems requiring methodical thinking and calculation.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a particular value or relationship. They allow you to create concise, symbolic representations of mathematical problems.

• Variables represent unknown or changeable elements, like \( y \) and \( f \) in our example.
• Operators such as addition, subtraction, multiplication, and division connect these variables and numbers.

In dealing with the perimeter of a rectangle, suppose it involves both feet and yards as units. By converting everything into a single unit and expressing it with an algebraic expression like \( 3y + f \), we succinctly capture the mathematical operation. Such expressions make it easy to perform algebraic manipulations and solve for unknowns. Understanding and creating mathematical expressions is a fundamental skill in algebra that facilitates problem-solving and opens doors to deeper mathematical exploration.