Problem 45 - Algebra A Combined Function

Question

Traffic signs are regulated by the Manual on Uniform Traffic Control Devices (MUTCD). According to this manual, if the sign below is placed on a freeway, its perimeter must be 144 inches. Also, its length must be 12 inches longer than its width. Find the dimensions of this sign.

Step-by-Step Solution

Answer

Width: 30 inches, Length: 42 inches.

Step 1: Understanding the Problem

We are asked to find the dimensions of a rectangular traffic sign where the perimeter is 144 inches, and its length is 12 inches more than its width.

Step 2: Expressing the Relationships as Equations

Let the width of the sign be \( w \) inches. According to the problem, the length \( l \) is 12 inches more than the width, so we have \( l = w + 12 \). The perimeter of a rectangle is given by \( 2l + 2w = 144 \).

Step 3: Substituting the Length Expression

Substitute \( l = w + 12 \) into the perimeter equation: \( 2(w + 12) + 2w = 144 \).

Step 4: Simplifying the Equation

Distribute and simplify the equation: \( 2w + 24 + 2w = 144 \). Combine like terms: \( 4w + 24 = 144 \).

Step 5: Solving for the Width

Isolate \( w \) by subtracting 24 from both sides: \( 4w = 120 \). Then divide both sides by 4: \( w = 30 \).

Step 6: Finding the Length

Substitute the value of \( w \) back into the equation for \( l \): \( l = w + 12 = 30 + 12 = 42 \).

Step 7: Conclusion

The dimensions of the sign are a width of 30 inches and a length of 42 inches.

Key Concepts

Rectangular Perimeter

The perimeter of a rectangle is a key concept in solving geometric problems. When you hear "perimeter," think about the total distance around the shape.
In a rectangle, the perimeter is calculated using the formula:

• \( P = 2l + 2w \)

This formula takes into account the two pairs of equal sides that make up a rectangle. For instance, if you have a rectangle with a length (\( l \)) and a width (\( w \)), you multiply each by 2 and add the results to find the perimeter. This is a fundamental concept in both geometry and algebra that helps us solve for unknown dimensions when given specific conditions.
In our exercise, knowing the perimeter allowed us to set up an equation to find the width and length of the traffic sign. Remember, whenever you're dealing with a perimeter problem, identify what you know about lengths and widths first to set up your expression correctly.

Algebraic Expressions

Algebraic expressions are like puzzles that represent numbers with symbols, primarily to make solving problems more systematic. In this exercise, expressing relationships using algebra helps to simplify what could otherwise be a challenging solution.

• For example, we expressed the length of the sign in relation to its width: \( l = w + 12 \).
• Then, the perimeter equation \( 2l + 2w = 144 \) incorporated these variables directly into a manageable algebraic form.

By substituting \( l \) with \( w + 12 \) into the perimeter expression, we turn a problem of two variables into a solvable equation with one variable. This substitution reduces complexity, allowing you to isolate and find the value of a single unknown—here, \( w \) (width). Substitution is a powerful algebraic technique, especially in geometry, when dimensions are interrelated.

Problem Solving Steps

Understanding the steps to solve an equation is crucial in finding the right solution. Let's break it down so it becomes second nature when you encounter similar problems.

• First, identify what you know. Here, we had the total perimeter and a relationship between length and width.
• Next, translate the verbal problem into algebraic expressions. Use a formula like the perimeter's \( 2l + 2w = 144 \).
• Substitute known values or expressions into your equation to reduce the variables.
• Simplify the equation by combining like terms and isolating the variable of interest.
• Solve the equation step-by-step, ensuring precision in calculations.
• Finally, substitute back to check the consistency and accuracy of the found values against the conditions given in the problem.

Following these comprehensive steps not only helps solve this specific problem but builds a strategy applicable in numerous math challenges.

Geometry

Geometry is the study of shapes, their properties, and their relationships in space. In this problem, understanding geometric principles about rectangles helps us conceptualize the relationship between width, length, and perimeter.
A rectangle has two sets of parallel sides with equal lengths. By understanding this, you see why calculating a perimeter involves doubling both the length and the width. Geometrical understanding goes hand in hand with algebra to resolve such problems.
Moreover, in geometry, dimensions have to satisfy given conditions physically, enhancing spatial reasoning. This exercise highlights practical applications of geometry in real-world contexts, such as designing traffic signs that comply with specific rules, thus entwining mathematical theory with practical utility.