Problem 276 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. A rectangular rug has perimeter 240 inches. The length is 12 inches more than twice the width. Find the length and width of the rug.

Step-by-Step Solution

Answer

The width is 36 inches, and the length is 84 inches.

Step 1 - Define variables

Let the width of the rug be denoted as \(w\) inches. The length of the rug, which is 12 inches more than twice the width, can be denoted as \(l = 2w + 12\) inches.

Step 2 - Use the perimeter formula for a rectangle

The perimeter \(P\) of a rectangle is given by the formula \[P = 2l + 2w\]. From the exercise, the perimeter is 240 inches, so: \[2l + 2w = 240\]

Step 3 - Substitute the length into the perimeter equation

Substitute \(l = 2w + 12\) into \[2l + 2w = 240\]. This results in: \[2(2w + 12) + 2w = 240\]

Step 4 - Simplify the equation

Expand and simplify: \[4w + 24 + 2w = 240\] \[6w + 24 = 240\] \[6w = 216\]

Step 5 - Solve for the width

Solve for \(w\) by dividing both sides by 6: \[w = 36\]

Step 6 - Calculate the length

Use the expression for the length: \[l = 2w + 12\] and substitute \(w = 36\). \[l = 2(36) + 12 = 72 + 12 = 84\]

Key Concepts

perimeter of a rectangle

The **perimeter of a rectangle** is the total distance around the edges of the rectangle. It's like walking around the boundary of the rectangle once. To find the perimeter, you can use the formula:

• **Perimeter** = 2*length + 2*width

In this formula, you multiply both the length and the width by 2, and then add the results together. This gives you the total length of all four sides.
In our exercise, the rug has a perimeter of 240 inches. We use this information to find the specific measurements of the length and the width later.

solving for width

To solve for the width, first we need to understand how to set up the problem. Let the width be designated as **w**.

The exercise tells us the length is 12 inches more than twice the width. This can be translated into an equation:

**length** = 2*width + 12

This equation represents how the length is related to the width.

Next, use the perimeter formula which involves both length and width:

• 240 = 2length + 2width

We substitute the equation for length into this perimeter equation to solve for the width.

algebraic substitution

Algebraic substitution is a method used to replace one variable with an equivalent expression. In this exercise, we substitute **length** (expressed in terms of width) into the perimeter equation.

• **Length** = 2*width + 12

This substitution into the perimeter formula gives us:
$$
2*(2w + 12) + 2w = 240
By substituting **2w+12** for length, we simplify solving for width.

simplifying algebraic expressions

Once we substitute the length with **2w + 12** in the perimeter equation, we have:

2*(2w + 12) + 2w = 240
The next step involves simplifying this expression. First, distribute the **2** inside the parentheses:

• 2 * 2w = 4w


• 2 * 12 = 24

So it becomes:

• 4w + 24 + 2w = 240

Combine the **w** terms:

• 4w + 2w = 6w

Therefore,

• 6w + 24 = 240
Finally, subtract **24** from both sides:
• 6w = 216
Then divide by **6**:
• w = 36

The width of the rug is 36 inches.