Problem 16 - Intermediate Algebra

Question

Use a perimeter formula to find the unknown measurement. See Example 1. Interior Decorating. Sixty-six feet of crown molding was installed around a rectangular dining room. If the length of the room is 18.75 feet, what is its width?

Step-by-Step Solution

Answer

The width of the dining room is 14.25 feet.

Step 1: Understand the Problem

We need to determine the width of a rectangular dining room given that the perimeter is 66 feet and the length is 18.75 feet. We can use the perimeter formula for a rectangle to find the width.

Step 2: Write the Perimeter Formula for a Rectangle

The formula for the perimeter of a rectangle is given by: \[ P = 2L + 2W \]where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width.

Step 3: Substitute the Known Values

We know the perimeter \( P = 66 \) feet and the length \( L = 18.75 \) feet. Substitute these values into the formula:\[ 66 = 2(18.75) + 2W \]

Step 4: Simplify the Equation

Calculate \( 2 \times 18.75 \):\[ 2 \times 18.75 = 37.5 \]Substitute back to get:\[ 66 = 37.5 + 2W \]

Step 5: Solve for Width \( W \)

Subtract 37.5 from both sides of the equation:\[ 66 - 37.5 = 2W \]\[ 28.5 = 2W \]Divide both sides by 2:\[ W = \frac{28.5}{2} \]\[ W = 14.25 \]

Step 6: Interpret the Result

The width of the dining room is 14.25 feet.

Key Concepts

Rectangle Dimensions

Rectangles are four-sided shapes that have opposite sides of equal length. When we talk about dimensions of a rectangle, we are referring to its length and width.
To visualize this, imagine a long dining room; the longest wall is the length and the shorter wall is the width. In our exercise, we know one of the dimensions, the length, which is 18.75 feet.

Understanding the dimensions is crucial because it allows us to apply mathematical formulas correctly. Armed with the rectangle's dimensions, you can find other attributes like area and perimeter. Knowing just the length or the width isn't enough to solve problems involving perimeter or area calculations. You usually need both.

• Length is generally considered the longer side.
• Width is the shorter side, often right opposite the length.

Knowing these measurements is foundational for solving problems involving rectangles.

Perimeter Formula

The perimeter of a shape is the total distance around its edges. For rectangles, this means adding up the lengths of all four sides.
The perimeter formula for a rectangle is given as:
\[ P = 2L + 2W \]where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width.

To calculate perimeter, you multiply each dimension (length and width) by 2 because these shapes have two lengths and two widths. You then sum these products together.

• This formula helps you determine either the length or width if the other dimension and perimeter are provided.
• It can also verify if an existing rectangle's measurements are correct if the perimeter is already known.

Therefore, having a firm grasp of this formula is key for rectangle-related problems.

Width Calculation

To find the width of a rectangle, especially when you're given the perimeter and length, you'll rely on rearranging the perimeter formula - an algebraic technique.
Here's how:
Start with the formula \( P = 2L + 2W \), and substitute any known values. From our exercise, \( P = 66 \) and \( L = 18.75 \).

Substitute these into the formula:
\[ 66 = 2(18.75) + 2W \]
Simplifying gives you:
\[ 66 = 37.5 + 2W \]

• Subtract 37.5 from 66 to isolate the \( 2W \) term.
• Then divide by 2 to solve for \( W \).

Complete these steps, and you'll find the width:
\[ W = 14.25 \]. This method effectively isolates the width, allowing you to solve for it swiftly when perimeter and length are known.