Problem 24 - Fundamentals of Mathematics

Question

The perimeter of a rectangle is 38 feet. Find the length and width of the rectangle if the length is 5 feet less than three times the width.

Step-by-Step Solution

Answer

The width is 6 feet and the length is 13 feet.

Step 1: Define Variables

Let's define the width of the rectangle as \( w \). According to the problem, the length is 5 feet less than three times the width. Therefore, we can express the length as \( l = 3w - 5 \).

Step 2: Write the Perimeter Formula

The formula for the perimeter \( P \) of a rectangle is given by \( P = 2l + 2w \). We know that the perimeter is 38 feet, so we can set up the equation: \( 38 = 2l + 2w \).

Step 3: Substitute the Expression for the Length

Substitute the expression for \( l \) from Step 1 into the perimeter equation: \( 38 = 2(3w - 5) + 2w \).

Step 4: Simplify the Equation

Distribute the 2 in the equation: \( 38 = 6w - 10 + 2w \). Combine like terms to simplify the equation: \( 38 = 8w - 10 \).

Step 5: Solve for the Width

Add 10 to both sides: \( 48 = 8w \). Then divide both sides by 8 to isolate \( w \): \( w = 6 \).

Step 6: Calculate the Length

Substitute the width \( w = 6 \) back into the expression for the length: \( l = 3(6) - 5 \). Calculate \( l = 18 - 5 \), so \( l = 13 \).

Step 7: Verify the Solution

Check if the found dimensions satisfy the perimeter equation: \( 2l + 2w = 38 \). Substitute \( l = 13 \) and \( w = 6 \), we get \( 2(13) + 2(6) = 26 + 12 = 38 \). The calculation is correct.

Key Concepts

Understanding Rectangle Dimensions

When working with a rectangle problem like this one, you first need to understand what dimensions are involved. A rectangle has two main measurements: length and width. In our problem, these dimensions are linked by a special condition: the length is 5 feet less than three times the width. This relationship is crucial because it helps us express the length in terms of the width. By using simple substitution, we can find both our dimensions in one set of calculations. This is a powerful technique in math that allows us to solve complex problems more easily.
To clarify, consider a visual image of a rectangle where you can adjust its sides based on given conditions. Understanding this concept helps you not just solve this mathematical problem, but also apply similar logic to different real-world scenarios.

Exploring Algebraic Expressions

Algebraic expressions can look intimidating, but they're simply another way of representing numbers using variables and constants. In our exercise, we defined the width of the rectangle with the variable \( w \). From the given condition about the length, we expressed it as an algebraic expression: \( l = 3w - 5 \).
This expression tells us how the length depends on the width. By assigning values to \( w \), we can easily calculate the length. Remember, an algebraic expression keeps transforming to reflect real values based on initial conditions. To become comfortable with such expressions, start by breaking them down into smaller parts and focus on how each part affects the other.

Applying the Perimeter Formula

The formula for the perimeter of a rectangle is simple yet fundamental in solving problems like this. It's given by:

• Perimeter \( P = 2l + 2w \)

In this situation, the perimeter is already known to be 38 feet. By substituting the expressions for length \( l \) and width \( w \), the formula becomes an equation we can solve. We replaced \( l \) with our expression from earlier, \( 3w - 5 \), and substituted it into the perimeter formula to form:

• \( 38 = 2(3w - 5) + 2w \)

This step is where an understanding of algebra helps, allowing us to break down each term and combine like terms, resulting in a manageable equation that gives us the width once solved. Understanding this formula is useful because it’s one of the many stepping stones to conquering mathematical problems with multiple steps.

Mathematical Verification

Verification is a key step in any math problem to ensure everything adds up correctly. Once you find the dimensions, it’s necessary to check your work, much like proofreading an essay. To verify, substitute the found dimensions back into the original perimeter equation to see if you arrive at the correct perimeter.
For our rectangle, the solution found the width \(w=6\) and the length \(l=13\). Plugging them back into the perimeter equation, we must find:

• \( 2(13) + 2(6) = 26 + 12 = 38 \)

This successful re-calculation confirms our dimensions are correct. Verification not only helps catch errors but also provides reassurance that our mathematical processes are sound. In real-world applications, this step could prevent costly mistakes, reaffirming the importance of double-checking your calculations.