Problem 19 - Elementary Algebra

Question

The length of a rectangle is four ninths its width. The area is 144 square feet. Find the dimensions.

Step-by-Step Solution

Answer

Answer: The dimensions of the rectangle are a length of 8 feet and a width of 18 feet.

Step 1: Define the variables

Let's call the length of the rectangle 'l' and the width 'w'.

Step 2: Write the given relationship as an equation

We are given that the length is 4/9 times the width. So, we can write: l = (4/9)w.

Step 3: Write the equation for the area of the rectangle

We are given the area 'A' of the rectangle is 144 square feet. The formula for the area of a rectangle is: A = lw. So, we have: 144 = lw.

Step 4: Substitute the relationship equation into the area equation

We will now use the relationship from Step 2 and substitute it into the area equation from Step 3: 144 = ((4/9)w)w.

Step 5: Solve for 'w'

To solve for 'w', we will first simplify the equation: 144 = (4/9)w^2. Now, multiply both sides by 9/4 to isolate w^2: (9/4) * 144 = w^2. Evaluating this gives us: 324 = w^2. Now, take the square root of both sides to find the width: w = sqrt(324) = 18.

Step 6: Find the length using the relationship equation

Now that we have found the width, we can use the equation from Step 2 to find the length: l = (4/9)w = (4/9)*(18) = 8.

Step 7: State the dimensions of the rectangle

After solving the system of equations, we find that the dimensions of the rectangle are: length = 8 feet and width = 18 feet.

Key Concepts

Rectangle Geometry

A rectangle is a quadrilateral with opposite sides being equal in length. It consists of four right angles, making it a type of parallelogram. Understanding the properties of rectangles is vital as it helps in solving problems such as finding dimensions when certain constraints like area or proportions are known.

Key features of rectangle geometry include:

• The opposite sides are parallel.
• All angles are 90 degrees.
• The diagonals are equal in length and bisect each other.

These properties form the foundation for calculations involving rectangles, especially when solving complex geometric algebra problems.

Area Calculation

Determining the area of a rectangle is straightforward with the formula:\[ A = l imes w \]where \( l \) represents the length and \( w \) is the width.

This fundamental equation helps us understand the space covered by the rectangle. In problems where the area and one dimension's ratio to another is given, like our exercise, understanding this formula is crucial. Simply put:

• Area gives us the product of length and width.
• Knowing the area helps in back-calculating either dimension when additional information is provided.

Hence, starting calculations with the area formula enables the derivation of unknown dimensions when certain constraints are present.

Algebra Problem Solving

Algebra plays a pivotal role in geometry problem solving by using equations to find unknown variables. Such an approach makes it easier to handle relationships and constraints, like those between length and width in a rectangle.

Using algebra to solve geometry problems involves:

• Identifying known relationships and data, e.g., given ratios.
• Replacing unknowns with variables.
• Creating equations based on given conditions, like area.
• Simplifying and solving the equations to extract unknown values.

In our problem, step-by-step equation manipulation is used to find the rectangle's width first, followed by utilizing established relationships to determine the length.

Length and Width Proportions

In this exercise, the problem revolves around understanding the proportions between the length and width of a rectangle, where the length is a fraction of the width. Specifically, these proportions provide a crucial relational expression:\[ l = \frac{4}{9}w \]This means the length is four-ninths of the width.

When dealing with proportions like these:

• Express one dimension in terms of another using ratios.
• Use the ratio to substitute values into equations like the area calculation.
• Solving the resulting equation can provide the unknown dimension.

Proportions assist greatly in simplifying the problem, focusing on fewer variables and extracting dimensions systematically, as evident from the highlighted exercise solution.