Problem 222 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. The perimeter of a rectangular toddler play area is 100 feet. The length is ten more than three times the width. Find the length and width of the play area.

Step-by-Step Solution

Answer

The width is 10 feet and the length is 40 feet.

Step 1: Define the Variables

Let the width of the play area be denoted by \( w \) and the length be denoted by \( l \).

Step 2: Set Up Equations Based on the Perimeter

The perimeter of a rectangle is given by \( 2l + 2w \). Since the perimeter is 100 feet, the equation is \( 2l + 2w = 100 \).

Step 3: Set Up the Length Equation

The length is described as being ten more than three times the width. This translates to \( l = 3w + 10 \).

Step 4: Substitute the Length Equation into the Perimeter Equation

Substitute \( l = 3w + 10 \) into \( 2l + 2w = 100 \): \[ 2(3w + 10) + 2w = 100 \]

Step 5: Simplify and Solve for the Width

Expand and simplify the equation: \[ 6w + 20 + 2w = 100 \ 8w + 20 = 100 \] Subtract 20 from both sides: \[ 8w = 80 \ w = 10 \] Thus, the width \( w \) is 10 feet.

Step 6: Solve for the Length

Use the width to find the length: \[ l = 3w + 10 \ l = 3(10) + 10 \ l = 30 + 10 = 40 \] Thus, the length \( l \) is 40 feet.

Key Concepts

rectangular perimeter

The perimeter of a rectangle is the total distance around the outer edge of the rectangle. To find the perimeter, you add up the lengths of all four sides. Since a rectangle has two pairs of equal sides, the formula for the perimeter is simplified to:

\( P = 2l + 2w \)

Where:

• \( P \) is the perimeter
• \( l \) is the length
• \( w \) is the width

In the given problem, the perimeter of the rectangular toddler play area is 100 feet. This fundamental understanding allows us to translate it into an equation, forming the basis for solving the system of equations later on.

solving algebraic equations

Algebraic equations are mathematical statements that show the equality of two expressions. To solve an equation means to find the value of the variable that makes the equation true. This usually involves a series of steps to simplify the equation. Here’s a general approach:

• Isolate the variable on one side of the equation
• Perform the same mathematical operations on both sides to maintain balance
• Simplify the expressions as much as possible

In this exercise, we work through two algebraic equations derived from the problem’s conditions:

1. The perimeter equation: \( 2l + 2w = 100 \)
2. The length equation: \( l = 3w + 10 \)
Using these equations, we solve for one variable at a time.

variable substitution

Variable substitution is a method used to solve systems of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. This can simplify the process of finding the values of the variables.

In this problem:

• We first solve the length equation \( l = 3w + 10 \) for \( l \).
• Then, we substitute this expression for \( l \) into the perimeter equation \( 2l + 2w = 100 \).

This substitution allows us to create an equation with only one variable (in this case, width \( w \)), which we can then solve step by step.

geometry applications

Applying geometric principles to algebra can help solve real-world problems, such as finding dimensions of shapes. Rectangle problems often involve finding missing lengths or widths when certain conditions about the perimeter or area are given. Geometry gives us the formulas and algebra allows us to solve for unknowns systematically.

In the given exercise, understanding the geometric property of the perimeter helps us set up the necessary equations. The provided condition that the length is ten more than three times the width forms another equation. Using algebra and substitution techniques, we find the dimensions of the rectangle. This approach can be applied to various geometric problems involving different shapes and conditions.

algebra problem-solving

Solving algebra problems typically follows a structured approach which involves:

• Identifying what you need to find
• Defining variables to represent the unknowns
• Writing equations based on given conditions
• Solving the equations using algebraic methods like substitution or elimination

In the provided exercise, we systematically defined variables for the width and length of the play area, set up an equation for the perimeter, and another for the length based on the problem conditions. We used substitution to solve for one variable first, making it easier to find the value of the other variable. This process of breaking down the problem into smaller manageable steps is key to solving any algebraic problem effectively.