Problem 15 - Elementary Algebra Within Reach

Question

In Exercises \(11-16\), use a system of linear equations to find the dimensions of the rectangle that meet the specified conditions. \(35.2\) meters The length is \(120 \%\) of the width.

Step-by-Step Solution

Answer

The dimensions of the rectangle are: length = 9.6 meters, width = 8 meters.

Step 1: Set up the equations

Create two equations from the given information. The first equation is from the formula for the perimeter of a rectangle, which is 2(length + width) = 35.2. The second equation is based on the information that the length is 120% of the width. This can be written as length = 1.2 * width.

Step 2: Solve the system of equations

Solve the second equation for length to obtain a relation that can be substituted into the first equation. Substitute length = 1.2*width into the first equation to obtain: 2(1.2*width + width) = 35.2. This simplifies to 2.2*width = 35.2 / 2 = 17.6.

Step 3: Find the width

Solve for width by dividing 17.6 by 2.2. This gives width = 8 meters.

Step 4: Find the length

Substitute width = 8 meters into length = 1.2*width. This gives length = 1.2 * 8 = 9.6 meters.

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle represents the total distance around the edge of the rectangle. It is calculated by adding together the lengths of all four sides. Since rectangles have opposite sides that are equal in length, the perimeter can be easily calculated with the formula:\[ P = 2 \times (l + w) \]where \( P \) is the perimeter, \( l \) is the length, and \( w \) is the width. By knowing the perimeter, you can often find other dimensions of the rectangle when additional information is provided. In our problem, the given perimeter was 35.2 meters.

Solving Equations

Solving equations involves finding the values for the variables that make the equation true. It often requires manipulating the equation using operations such as addition, subtraction, multiplication, or division. When you solve equations, your goal is to isolate the variable on one side of the equation.

In the context of our rectangle problem, we are given the perimeter equation and an expression for the length. Substituting the value or expression of one variable in another equation can help us solve systems of equations by finding one or more unknown values.

Algebraic Substitution

Algebraic substitution is a method used to solve systems of equations. It involves replacing a variable in one equation with the expression for that variable from another equation. This helps reduce the number of variables, making the equation easier to solve.
- From the rectangle problem, we start with two initial equations: - Perimeter equation: \( 2(l + w) = 35.2 \) - Length as a percentage of width: \( l = 1.2 \times w \)By substituting the expression for \( l \) from the length equation into the perimeter equation, you can solve for \( w \), the width. Once the width is found, you can substitute back to find the length using the equation \( l = 1.2 \times w \).

Problem Solving in Algebra

Problem solving in algebra often involves a series of logical steps designed to find unknown quantities. Here are some key strategies we used:

• Understand the problem: Identify what is known and what needs to be found.
• Formulate equations: Translate the problem's conditions into equations.
• Use substitution: Simplify the system of equations by expressing one variable in terms of another.
• Carry out calculations: Use basic arithmetic operations to solve the equations.

In this problem, understanding the relationship between length and width and translating that into correct equations was crucial. By logically applying these steps, algebraic problems can be solved effectively.