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Chapter 2: Problem 30

A standard rectangular highway billboard sign has a perimeter of 124 ft. The length is 6 ft more than three times the width. Find the dimensions of the sign. (IMAGE CAN'T COPY)

Short Answer

Expert verified
Width = 14 ft, Length = 48 ft.

Step by step solution

01

Define variables

Let the width of the billboard be denoted by \(W\). According to the problem, the length is 6 ft more than three times the width. Let the length be denoted by \(L\). Thus, \[L = 3W + 6.\]
02

Set up the perimeter equation

The perimeter of a rectangle is given by the formula \[P = 2L + 2W.\] The problem states that the perimeter is 124 ft. Substituting the values, we get \[2L + 2W = 124.\]
03

Substitute the expression for L

From Step 1, we know that \[L = 3W + 6.\] Substitute this expression into the perimeter equation: \[2(3W + 6) + 2W = 124.\]
04

Simplify the equation

Distribute and combine like terms: \[6W + 12 + 2W = 124.\] This simplifies to \[8W + 12 = 124.\]
05

Solve for W

Isolate \(W\) by subtracting 12 from both sides: \[8W = 112.\] Then divide by 8: \[W = 14.\]
06

Find the length L

Use the value of \(W\) to find \(L\): \[L = 3W + 6 = 3(14) + 6 = 42 + 6 = 48.\]
07

Verify the solution

Check the perimeter with the dimensions found: \[2L + 2W = 2(48) + 2(14) = 96 + 28 = 124.\] The dimensions satisfy the perimeter requirement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rectangular perimeter
The perimeter of a rectangle refers to the total distance around the edge of the shape. It's important to understand this simple geometric concept, as it frequently appears in algebra problems. The formula to find the perimeter \( P \) of a rectangle is given by \[ P = 2L + 2W \] where \( L \) is the length and \( W \) is the width of the rectangle. In our example, the perimeter of the highway billboard is 124 ft. Knowing this, we can plug it into our equation to solve for the unknown dimensions. By setting up the equation \( 2L + 2W = 124 \), we take a significant step towards finding the exact measurements of the length and width.
variable definition
Defining variables is a primary and crucial step in solving algebraic problems. In this exercise, we start by assigning symbols to the unknown quantities:
  • Width \( W \)
  • Length \( L \)
The problem tells us that the length is 6 feet more than three times the width. This relationship can be expressed as an equation: \[ L = 3W + 6 \] By defining these variables clearly, we simplify the problem and create a structured approach to finding the solution.
equation simplification
Equation simplification involves reducing complex equations into simpler forms to make them easier to solve. After substituting \( L \) with \( 3W + 6 \) in our perimeter equation, the equation \( 2L + 2W = 124 \) becomes: \[ 2(3W + 6) + 2W = 124 \] To simplify, first distribute and combine like terms:
  • Distribute: \( 6W + 12 + 2W = 124 \)
  • Combine: \( 8W + 12 = 124 \)
Then, isolate \( W \) by performing the following steps:
  • Subtract 12 from both sides: \( 8W = 112 \)
  • Divide by 8: \( W = 14 \)
Thus, the width \( W \) is 14 ft. Using this, we can easily find the length \( L \) by substituting back into the equation \( L = 3W + 6 \), resulting in: \[ L = 3(14) + 6 = 48 \] Therefore, the dimensions of the billboard are 48 ft in length and 14 ft in width.

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