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Chapter 1: Problem 28

The length of a rectangle is 3 feet more than its width. If the perimeter is 46 feet, then what is the width?

Short Answer

Expert verified
The width is 10 feet.

Step by step solution

01

- Understand the Problem

The problem states that the length (L) of a rectangle is 3 feet more than its width (W). The perimeter of the rectangle is given as 46 feet. We need to find the width.
02

- Write Down the Formula

The formula for the perimeter (P) of a rectangle is given by \[ P = 2 \times (L + W) \] Given that the perimeter (P) is 46 feet, we substitute P in the formula.
03

- Express Length in Terms of Width

Since the length (L) is 3 feet more than the width (W), we can write \[ L = W + 3 \]
04

- Substitute L in the Perimeter Formula

Substitute \( L = W + 3 \) in the perimeter formula: \[ 46 = 2 \times ((W + 3) + W) \]
05

- Simplify the Equation

Simplify the equation: \[ 46 = 2 \times (2W + 3) \] Then \[ 46 = 4W + 6 \]
06

- Solve for Width

Subtract 6 from both sides of the equation: \[ 40 = 4W \] Divide both sides by 4: \[ W = 10 \]
07

- Verify the Solution

Verify the width by calculating the length and perimeter: \[ L = W + 3 = 10 + 3 = 13 \] Perimeter = 2 \times (L + W) = 2 \times (10 + 13) = 46 feet, which matches the given data.

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Key Concepts

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

Perimeter of a Rectangle
In geometry, the perimeter of a rectangle is the total distance around the outside of the rectangle. It can be calculated using the formula: \[ P = 2 \times (L + W) \] Where:
  • \( P \) is the perimeter
  • \( L \) is the length
  • \( W \) is the width
In the given problem, the perimeter is 46 feet. Using this formula, we can substitute the known values to find the unknown width of the rectangle. This formula is essential for solving any problems related to the perimeter of geometric shapes.
System of Equations
A system of equations consists of two or more equations that have common variables. To solve a system of equations, you need to find the values of the variables that satisfy all equations in the system. In this problem, we have two main equations:
  • \[ 46 = 2 \times (L + W) \] (perimeter formula)
  • \[ L = W + 3 \] (length is 3 feet more than width)
These two equations together form a system of equations. Our goal is to solve for both \( L \) (length) and \( W \) (width) of the rectangle. Understanding how to work with systems of equations is a crucial skill in algebra since it applies to many different types of math problems.
Basic Algebraic Manipulation
Basic algebraic manipulation involves operations like addition, subtraction, multiplication, and division to rearrange and solve equations. This problem requires several steps of manipulation:
  • First, express length \( L \) in terms of width \( W \): \[ L = W + 3 \]
  • Next, substitute \( L \) into the perimeter formula: \[ 46 = 2 \times ((W + 3) + W) \]
  • Simplify the equation: \[ 46 = 2 \times (2W + 3) \]
  • Simplify further and solve for \( W \): \[ 46 = 4W + 6 \rightarrow 40 = 4W \rightarrow W = 10 \]
Being able to simplify and solve linear equations is a fundamental skill in algebra that enables you to solve more complex problems.
Substitution Method
The substitution method is a common technique for solving systems of equations. Here's how it works:
  • Solve one of the equations for one variable.
  • Substitute this expression into the other equation.
  • Solve the second equation for the remaining variable.
  • Substitute the solution back into the first equation to find the other variable's value.
In this problem, we substituted \( L = W + 3 \) (from the length equation) into the perimeter formula: \[ 46 = 2 \times ((W + 3) + W) \]By simplifying and solving, we found \( W = 10 \). We then substituted \( W = 10 \) back into \( L = W + 3 \) to get \( L = 13 \). The substitution method is very useful for solving dual-variable linear equations effectively.

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