Problem 20 - Algebra 2 and Trigonometry

Question

The ratio of the length to the width of a rectangle is 5 : 4. The perimeter of the rectangle is 72 inches. What are the dimensions of the rectangle?

Step-by-Step Solution

Answer

The width is 16 inches and the length is 20 inches.

Step 1: Understand the Problem

To solve for the dimensions of the rectangle, we need to focus on the given ratio of length to width, which is 5:4, and the perimeter, which is 72 inches. We'll use these to find the actual dimensions.

Step 2: Set Up Equations Based on the Ratio

Let's denote the width of the rectangle as \(4x\) and the length as \(5x\), based on the ratio 5:4. This means, if the width is 4 times some number \(x\), the length is 5 times that same number.

Step 3: Use the Perimeter Formula

The perimeter of a rectangle is given by the formula \(P = 2L + 2W\). Substitute the expressions for length and width: \(2(5x) + 2(4x) = 72\).

Step 4: Simplify and Solve the Equation

Simplify the equation: \(10x + 8x = 72\). Combine like terms to get \(18x = 72\). Divide both sides by 18 to solve for \(x\): \(x = 4\).

Step 5: Find the Dimensions

Now that we have \(x = 4\), substitute back to find the dimensions. The width is \(4x = 16\) inches and the length is \(5x = 20\) inches.

Key Concepts

Ratio of Length to Width

When discussing rectangles, the **ratio of length to width** is a helpful tool for understanding how the dimensions relate to each other. In this problem, the ratio is given as 5:4. This means for every 5 units of length, there are 4 units of width. Ratios are powerful because they allow us to express one quantity in terms of another using simple numbers, highlighting proportions without needing precise measurements upfront.

To work with this ratio in a practical way, assume a variable, often denoted as \(x\), which represents a common multiplier for the ratio parts. For this rectangle, it translates to the equations:

• The length is represented as \(5x\).
• The width is represented as \(4x\).

These expressions maintain the 5:4 relationship while letting us solve for specific dimensions once more data, like the perimeter, is factored in.

Perimeter Formula

The **perimeter of a rectangle** is the total distance around it. For rectangles, the formula is straightforward:

\[ P = 2L + 2W \]

This equation adds up the lengths of all sides, with \(L\) being the length and \(W\) being the width. Here, each dimension is doubled because opposite sides of a rectangle are equal.

When we use this formula in context, we utilize the expressions for length and width derived from the ratio (i.e., \(L = 5x\) and \(W = 4x\)), resulting in:

\[ 2(5x) + 2(4x) = P \]

This simplifies to:

\[ 10x + 8x = P \]

Since the perimeter is given as 72 inches, inserting this into the equation is the next crucial step to finding specific dimensions.

Solving Linear Equations

Once we have our equation set up from the perimeter formula, we enter the world of **solving linear equations**. These equations contain variables whose highest power is one, making them linear. In this case, we already derived the equation:

• \(10x + 8x = 72\)

Combining like terms simplifies the equation to:

• \(18x = 72\)

Solving for \(x\) requires isolating the variable on one side of the equation. We do this by dividing each side by 18, yielding:

• \(x = 4\)

With the value of \(x\) known, we can calculate the rectangle's dimensions by substituting back into our width and length equations:

• Width: \(4x = 16\) inches
• Length: \(5x = 20\) inches

Solving linear equations solidifies the concrete dimensions from abstract relationships, completing the rectangle's size puzzle.