Problem 7 - Prealgebra

Question

GEOMETRY The perimeter of a rectangle is 20 feet. The width is 4 feet less than the length. Find the dimensions of the rectangle.

Step-by-Step Solution

Answer

The length is 7 feet and the width is 3 feet.

Step 1: Understanding the Perimeter Formula

The perimeter of a rectangle is calculated using the formula: \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width of the rectangle. In this problem, we know the perimeter is 20 feet.

Step 2: Formulate the Width Expression

We are told that the width is 4 feet less than the length. We can express this relationship as \( w = l - 4 \).

Step 3: Substitute Width in Perimeter Equation

Substitute \( w = l - 4 \) into the perimeter formula: \( 20 = 2(l + (l - 4)) \). Simplify the equation: \( 20 = 2(2l - 4) \).

Step 4: Solve for Length

Continue to simplify the equation: \( 20 = 4l - 8 \). Add 8 to both sides to get \( 28 = 4l \). Divide both sides by 4 to isolate \( l \): \( l = 7 \).

Step 5: Find the Width

Substitute \( l = 7 \) back into the expression for width: \( w = l - 4 = 7 - 4 = 3 \).

Step 6: Verify the Solution

Check that these dimensions satisfy the original perimeter condition: \( P = 2(l + w) = 2(7 + 3) = 20 \). The dimensions are correct.

Key Concepts

Rectangle Perimeter

In geometry, understanding how to calculate the perimeter of a rectangle is fundamental.
The perimeter is essentially the total distance around the edge of the rectangle. It's like if you were to walk along the boundary of a rectangular garden.
The formula to find the perimeter is given by \( P = 2(l + w) \), where \( P \) denotes the perimeter, \( l \) is the length, and \( w \) is the width. This equation simply adds up all the sides twice, as a rectangle has two lengths and two widths.
For instance, if you know one measurement of the perimeter, you can deduce missing dimensions. In our example, knowing the perimeter is 20 feet helps us ultimately find both length and width by setting up equations accordingly.
Mastering this formula allows one to tackle similar geometry problems with ease.

Algebraic Expressions

Algebraic expressions are a way to describe mathematical concepts using numbers, variables, and operations.
In the realm of problem-solving, like with rectangles, they help translate words into math. An expression such as \( w = l - 4 \) connects the width and length of the rectangle using algebra.
Expressions are powerful because they allow students to manipulate and solve problems using logical operations. In the provided solution, substituting expressions into the perimeter formula gives us a solvable equation involving only one variable.
Practice forming and using expressions helps in understanding how different elements in a math problem are related. This step is crucial before diving into more advanced topics in algebra.

Problem Solving in Mathematics

Problem solving is a key skill in mathematics. It often involves breaking down a complex problem into manageable steps. This exercise demonstrates a structured approach:

• First, understand the problem, identifying given information and what needs to be calculated.
• Next, translate the problem into mathematical expressions and equations. Use relationships between known and unknown quantities.
• Solve the equations systematically step-by-step, paying attention to operations and logical sequence.
• Finally, verify the solution by checking if it meets the original conditions of the problem.

Each step reinforces logical reasoning and mathematical skills, preparing one to apply this methodology to a variety of mathematical contexts.
Cultivating problem-solving abilities is essential for success in both academic and real-world scenarios.