Problem 37 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The length of a rectangle is 3 feet less than twice its width. If the perimeter is 54 feet, find the dimensions of the rectangle.

Step-by-Step Solution

Answer

The width is 10 feet and the length is 17 feet.

Step 1: Understanding the Problem

We are given a rectangle where the length is 3 feet less than twice the width, and the perimeter is 54 feet. We need to find the dimensions, specifically the width and length.

Step 2: Defining Variables

Let \( w \) represent the width of the rectangle. According to the problem, the length \( l \) can be described as \( l = 2w - 3 \).

Step 3: Writing the Perimeter Formula

The formula for the perimeter \( P \) of a rectangle is given by \( P = 2l + 2w \). We know \( P = 54 \), so we write the equation as \( 54 = 2l + 2w \).

Step 4: Substituting the Length Expression

Substitute the expression for \( l \) in terms of \( w \) into the perimeter equation: \( 54 = 2(2w - 3) + 2w \).

Step 5: Solving the Equation for Width

Simplify and solve the equation:1. Expand: \( 54 = 4w - 6 + 2w \)2. Simplify: \( 54 = 6w - 6 \)3. Add 6 to both sides: \( 60 = 6w \)4. Divide by 6: \( w = 10 \). So, the width is 10 feet.

Step 6: Finding the Length

Substitute \( w = 10 \) back into the expression for length: \( l = 2(10) - 3 = 20 - 3 = 17 \). So, the length is 17 feet.

Key Concepts

Rectangle Geometry Simplified

In geometry, a rectangle is a four-sided shape known as a quadrilateral. A key aspect of rectangles that makes them unique is their right angles. Each internal angle measures exactly 90 degrees. This property helps ensure that opposite sides of a rectangle are both equal and parallel.
Additionally, here are some vital attributes of rectangles to remember:

• They have two pairs of parallel sides.
• The longer side is referred to as the 'length'.
• The shorter side is referred to as the 'width'.
• Every rectangle is symmetrical along its central axis.

Understanding these properties helps us relate the dimensions of a rectangle to mathematical equations, allowing for the computation of unknown measurements like length and width. Recognizing these basic properties lays a fundamental groundwork for dealing with more complex geometric problems.

The Essentials of Perimeter Calculation

The perimeter of a shape is the total distance around it. For rectangles, calculating the perimeter involves adding up the lengths of all sides. Since opposite sides of a rectangle are equal, the formula for perimeter becomes straightforward:

• The formula is: \( P = 2l + 2w \).
• \( l \) represents the length, and \( w \) symbolizes the width.
  

For example, if you know the perimeter and one of the dimensions, you can easily find the missing dimension. This formula is a critical part of solving any geometry problem involving perimeter. By re-arranging and inserting known values into this formula, you can arrive at solutions with ease. This straightforward relationship between the formula components allows students to connect physical dimensions to their algebraic representations efficiently.

Mastering Solving Equations

Algebraic equations are expressions that demonstrate the equality of two mathematical statements using equals signs (=). Solving these equations involves finding the unknown value, often termed a variable. To solve algebraic equations efficiently:

• First, isolate the variable on one side of the equation.
• Use inverse operations to simplify each step, such as addition, subtraction, multiplication, or division.
  

In our exercise, for instance, we used the relationship between the length and width of a rectangle to form an equation. From there, it involves substitution and balancing calculations to unveil the unknown dimensions. The process might look like rearranging terms and solving step-by-step, streamlining towards uncovering the correct sides' measurements. Mastering such techniques tips the balance from simple problems to just about any level of algebraic complexity.