Problem 13 - 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. 16 yards The width is one-third of the length.

Step-by-Step Solution

Answer

The dimensions of the rectangle are length \(9.6\) yards and width \(3.2\) yards.

Step 1: Set up the system of equations

The given condition can be translated into equations. It stated that:1. The perimeter is 16 yards, so we can set up this equation: \(2 \times (length + width) = 16\).2. The width is one-third of the length, which gives us the second equation: \(width = 1/3 \times length\). Now, we have a system of two equations with two variables, namely length and width.

Step 2: Substitute the second equation into the first

We can substitute the second equation into the first to solve for one variable. So replacing width in the first equation with \(1/3 \times length\) results in: \(2 \times (length + 1/3 \times length) = 16\). Simplifying the equation, we get \(5/3 \times length = 16\).

Step 3: Solve for the length

To solve for length, divide both sides of the equation by \(5/3\). This gives length equals 16 divided by \(5/3\), or \(48/5\), which simplifies to \(9.6\) yards.

Step 4: Solve for the width

Once length is found, substitute it into the second equation to find width. So, substituting \(length = 9.6\) into \(width = 1/3 \times length\) gives width equals \(1/3 \times 9.6\), or \(3.2\) yards.

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the outside of the rectangle. Imagine if you were to walk all the way around it, you'd measure the perimeter. It's calculated by adding the lengths of all four sides. If you know just the length and the width of the rectangle, you can use the formula \( P = 2 \times (\text{length} + \text{width}) \) to find the perimeter.
In many math problems, like the one we are discussing, you'll know the perimeter, and need to find the dimensions of the rectangle. This involves working backwards using the perimeter formula.
Remember:

• "\( 2 \times \text{length} \)" is for the two longer sides.
• "\( 2 \times \text{width} \)" is for the two shorter sides.

By learning to manipulate this formula, we can solve for missing dimensions when given the perimeter.

Solving Equations

Equations are like puzzles; they show relationships between numbers using mathematical symbols. When you solve an equation, you're figuring out the value of an unknown variable.
In our rectangle problem, we have two equations:

• One describing the perimeter: \( 2 \times (\text{length} + \text{width}) = 16 \).
• Another relating width and length: \( \text{width} = \frac{1}{3} \times \text{length} \).

The key to solving this system is substitution: replacing one variable based on another equation.
This involves taking the second equation and substituting it into the first. It simplifies our job by allowing us to solve for one variable first and then find the other easily afterward.
Mastering these steps will help you tackle any linear equation system with confidence.

Length and Width Relationships

In mathematics, understanding the relationship between length and width in rectangles is crucial, especially when they give you conditions like 'width is one-third of length.' This direct relationship helps set up equations intuitively.
Say you're given that the width \( W \) is one-third the length \( L \). This relationship is expressed as \( W = \frac{1}{3} L \). This simple equation allows us to express one variable in terms of another, making calculations simpler.
When we substitute this relationship into the perimeter equation \( 2(L + W) = 16 \), it helps us find one of the dimensions directly. Once we have the length, finding the width becomes straightforward using our initial relationship.
These understanding ensures that even with limited information, you can deduce the dimensions of a rectangle accurately, demonstrating the power of algebra and relationships between dimensions.