Problem 97 - Intermediate Algebra

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

We need to convert the perimeter given in yards and feet into only feet.

Step 2: Conversion from Yards to Feet

Recognize that 1 yard is equal to 3 feet. So, if the perimeter is given as \(y\) yards, in terms of feet, it is \(3y\) feet.

Step 3: Combine Yard Conversion with Feet

The total perimeter in feet will be a combination of the converted yards and the given feet. Thus, the expression becomes \(3y + f\), where \(3y\) is the yard portion in feet and \(f\) is the feet portion.

Key Concepts

Unit Conversion

In many mathematical scenarios, we encounter situations where we need to convert units to allow for proper calculations or comparisons. This is known as unit conversion. For example, in this exercise, the perimeter is given in yards and feet, but we need it only in feet. Knowing that 1 yard equals 3 feet is crucial. This conversion factor is a constant that helps you transform one unit into another accurately. If you have a measurement in yards, simply multiply the number of yards by 3 to convert it into feet. This way, you can ensure consistency across your calculations. Always keep a note of common conversion factors—they can be a lifesaver in solving complex problems!

Perimeter Calculation

Perimeter calculation involves figuring out the total distance around a geometrical shape. A rectangle, for example, has four sides with opposite sides being equal in length. The standard formula for perimeter, if the length is denoted by \(L\) and the width by \(W\), is \(2L + 2W\). For this exercise, we're given the perimeter using different units which requires conversion before calculation. Calculating a perimeter that involves multiple units first requires all measurements to be in the same unit. Once converted, as demonstrated, you sum up the values. In this task, we move from mixed units (yards and feet) to a single unit (feet) by using our understanding of how yards translate into feet.

Step-by-step Problem Solving

Solving mathematical problems step-by-step not only simplifies the process but also enhances understanding. Breaking down complex problems into smaller, manageable steps provides clarity and direction.

• **Identify and understand the problem:** Recognize what is being asked. We needed to convert a mixed measurement into a singular unit.
• **Convert units as necessary:** Knowing the conversion factor (1 yard = 3 feet) allows us to switch from yards to feet decisively.
• **Apply the conversion and combine results:** As shown in the solution, the yard measurement translated to feet is summed with the given feet to form a complete expression, \(3y + f\).

By thoroughly going through each step without haste, potential mistakes are minimized, leading to a reliable solution. Designing your problem-solving method in steps empowers you with a robust strategy applicable to various mathematical challenges.