Problem 72 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following problems algebraically. Be sure to label what the variable represents. The width of a rectangle is 4 less than \(\frac{2}{3}\) its length. If the perimeter of the rectangle is 3 times the length, find its dimensions.

Step-by-Step Solution

Answer

Length = 24 units, Width = 12 units.

Step 1 - Define the Variables

Let the length of the rectangle be denoted as \(L\) and the width be denoted as \(W\).

Step 2 - Express the Width in Terms of Length

According to the problem, the width \(W\) is 4 less than \(\frac{2}{3}\) of its length. This can be written as: \[ W = \frac{2}{3}L - 4 \]

Step 3 - Write the Perimeter Equation

The perimeter of a rectangle is given by the formula: \[ P = 2L + 2W \] The problem states that the perimeter is 3 times the length, so: \[ P = 3L \]

Step 4 - Set Up the Perimeter Equation

Substitute the perimeter from the given condition into the perimeter formula: \[ 3L = 2L + 2W \]

Step 5 - Substitute W in Terms of L

Substitute \( W = \frac{2}{3}L - 4 \) into the perimeter equation: \[ 3L = 2L + 2(\frac{2}{3}L - 4) \]

Step 6 - Simplify the Equation

Simplify the expression: \[ 3L = 2L + \frac{4}{3}L - 8 \] Combine like terms: \[ 3L = 2L + \frac{4}{3}L - 8 \] Subtract \(2L\) from both sides: \[ L = \frac{4}{3}L - 8 \] Multiply both sides by 3 to eliminate the fraction: \[ 3L = 4L - 24 \]

Step 7 - Solve for L

Rearrange the equation to solve for \(L\): \[ 3L - 4L = -24 \] \[ -L = -24 \] Divide by -1: \[ L = 24 \]

Step 8 - Find the Width

Substitute \( L = 24 \) back into the equation for \(W\): \[ W = \frac{2}{3}(24) - 4 \] \[ W = 16 - 4 \] \[ W = 12 \]

Step 9 - State the Dimensions

The dimensions of the rectangle are: Length \( L = 24 \) units Width \( W = 12 \) units

Key Concepts

Rectangle Dimensions

When talking about rectangles, we focus on two major dimensions: length and width.

• Length (often denoted by L) refers to the longer side of the rectangle.
• Width (often denoted by W) refers to the shorter side of the rectangle.

In our exercise, we need to express the width in terms of the length. According to the problem, the width is 4 units less than \( \frac{2}{3} \) of its length: \[ W = \frac{2}{3}L - 4 \]
This equation tells us how to calculate the width if we know the length. Knowing these dimensions allows us to perform further calculations, like finding the perimeter.

Perimeter Formula

The perimeter of a rectangle is the total distance around the outside edges. It's calculated using the formula:
\[ P = 2L + 2W \]

Per the exercise, the perimeter is 3 times the length of the rectangle, which translates to:
\[ P = 3L \]
Now, by substituting the given condition into the formula, we set up the perimeter equation:
\[ 3L = 2L + 2W \]
This equation allows us to solve for one dimension when the other is known. It bridges the relationship between length and width through perimeter.

Variable Substitution

To solve the equation, we must replace terms with their equivalent expressions from the problem. This step is known as variable substitution.

Here, the width (W) is expressed in terms of length (L): \[ W = \frac{2}{3}L - 4 \]
We substitute this into the perimeter equation:
\[ 3L = 2L + 2(\frac{2}{3}L - 4) \]
This substitution simplifies the perimeter equation to a form involving only one variable: L. It’s a crucial step in the problem-solving process, enabling us to isolate and solve for the unknown dimension.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the unknown variable. Let’s break down each step:

1. Distribute the 2 in the perimeter equation: \[ 3L = 2L + \frac{4}{3}L - 8 \]
2. Combine like terms and isolate L: \[ 3L - 2L = \frac{4}{3}L - 8 \] which simplifies to \[ L = \frac{4}{3}L - 8 \]
3. Clear the fraction by multiplying everything by 3: \[ 3L = 4L - 24 \]
4. Isolate L by subtracting 4L from both sides: \[ -L = -24 \]
5. Solve for L: \[ L = 24 \]
Finally, we find the width by substituting back into the width formula: \[ W = \frac{2}{3}(24) - 4 = 16 - 4 = 12 \]
Thus, the dimensions of the rectangle are length 24 units and width 12 units.