Problem 14 - Precalculus Mathematics for Calculus

Question

Express the given quantity in terms of the indicated variable. The perimeter (in \(\mathrm{cm}\) ) of a rectangle that is \(5 \mathrm{cm}\) longer than it is wide; \(w=\) width of the rectangle (in cm )

Step-by-Step Solution

Answer

The perimeter is expressed as \( 4w + 10 \).

Step 1: Understanding the Expression

First, it's important to understand that the perimeter of a rectangle is calculated as the sum of all its sides. For a rectangle, this can be expressed as: \[ P = 2 imes ( ext{length} + ext{width}) \] Given that the length is 5 cm longer than the width, we can write it as: \[ ext{length} = w + 5 \] where \( w \) is the width.

Step 2: Substitute the Length

Substitute the expression for the length into the perimeter formula. Replace \( \text{length} \) with \( w + 5 \) in the perimeter equation:\[ P = 2 imes ((w + 5) + w) \] This will allow us to express the perimeter solely in terms of the width \( w \).

Step 3: Simplify the Expression

Simplify the expression inside the parenthesis first:\[ P = 2 imes (2w + 5) \]Now distribute the multiplication:\[ P = 2 imes 2w + 2 imes 5 = 4w + 10 \] Thus, the perimeter expressed in terms of \( w \) is \( 4w + 10 \).

Key Concepts

Understanding Rectangles

Rectangles are fundamental shapes in geometry. They have four sides and four right angles. Opposite sides are equal in length. This unique property makes them easy to study and work with in math problems. By understanding rectangles, you can easily calculate perimeter and area.
In the context of our exercise, we need to find the perimeter, which refers to the total distance around the rectangle. It's important to remember that the perimeter of a rectangle can be calculated by adding up the lengths of all its sides. Alternatively, you can use the formula:

• Perimeter, \( P = 2 \times (\text{length} + \text{width}) \).

This equation will help us solve the problem efficiently by plugging in known values.

Width and Length Relationship in a Rectangle

In this exercise, understanding how the width and length relate to each other is crucial to solving the problem. We are given that the length of the rectangle is 5 cm longer than its width. This means that whenever you determine the width, you simply add 5 cm to find the length.
To express this algebraically:

• \( \text{length} = w + 5 \)

where \( w \) stands for the width. This relationship helps us to substitute the length in the perimeter formula, making calculations easier. By understanding this relationship, we can rearrange and manipulate equations effectively.

Crafting an Algebraic Expression

To solve the problem efficiently, it's essential to convert the given information into an algebraic expression. Algebraic expressions make problems more tractable by allowing replacements and clear computations.
Using the formula for perimeter and our known relationship between width and length, we have:

• \( P = 2 \times ((w + 5) + w) \)

This equation uses the width and the derived expression for the length, enabling us to keep the expression simple and in terms of a single variable, \( w \). Such expressions are powerful in problem-solving because they streamline the calculation process.

Approach to Problem Solving

Problem-solving in mathematics often involves breaking down complex problems into manageable parts. For our task, we took these steps:

• Identified what we need to find – the perimeter in terms of width.
• Understood the width-length relationship – expressed length as \( w + 5 \).
• Substituted this relationship into the perimeter formula.

By understanding each component and how they interact, we simplified our task and made it more approachable. Problem-solving isn't just about finding the answer; it's about understanding the "why" and "how" that lead to the solution.

Using Mathematical Formulas Effectively

Mathematical formulas are concise tools that help streamline calculations and find solutions efficiently. In geometry, such formulas enable accurate evaluation of quantities like perimeter, area, and volume.
In this example:

• We started with the perimeter formula \( P = 2 \times (\text{length} + \text{width}) \).
• Substituted the known relationship to get \( P = 2 \times (2w + 5) \).
• Simplified this to \( P = 4w + 10 \).

By following these steps and ensuring each substitution is done correctly, you can confidently arrive at the solution. Formulas are not just about memorization; they are about understanding how to apply them to find solutions effectively.