The **aim** of this question is to understand the **concepts** of the geometry of the **rectangle** and to understand the **formulas** to calculate the **area** and the **perimeter** of the rectangle.

According to **Euclidean** plane geometry, a rectangle is a **quadrilateral** with sides having all the **internal** angles equal to $90$ degrees. The **right** angle is **produced** when two sides **meet** at any corner. **Opposite** sides are equal in **length** in a rectangle, making it **different** from the **square** where all the four sides are **equal.**

The area is the amount that **represents** the size of a **region** on the plane or on a **curved** surface. The area of a **rectangle** is properly calculated by multiplying its **length** by **width.** Mathematically:

\[ A= Length \times Width \]

The **perimeter** of any **2D shape** can be calculated by adding the **length** of all of its sides. In a rectangle, the **perimeter** is computed by **adding** all four sides. Because the **opposites** sides are **equal** in length, the **formula** for the perimeter is:

\[ P = 2L + 2W \]

## Expert Answer

Given information:

Area of the **rectangular** banner: $A = 144 ft^2$

The **width** of the banner is $\dfrac{1} {4}$ the **length** of the banner: $ Width = \dfrac{Length} {4}$.

The **formula** for the area of a **rectangle **is**:**

\[ A = L \times W \]

Inserting the **Area** $A$.

\[ 144= L \times W \]

Now **inserting** $W = \dfrac{L} {4}$

\[ 144= L \times \dfrac{L} {4} \]

\[ 144= \dfrac{L^2} {4} \]

\[ L^2 = 144 \times 4 \]

\[ L^2 = 576 \]

Taking the **square** root on both **sides:**

\[ \sqrt{L^2} = \sqrt{576} \]

\[ L = \sqrt{576} \]

**Length** comes out to be:

\[ L = 24 ft \]

Now **find** the Width $W$ of the banner.

\[ W = \dfrac{L} {4} \]

Inserting $L = 24$:

\[ W = \dfrac{24} {4} \]

\[ W = 6 \]

## Numerical Answer

The **dimensions** of the banner is as follows: **Length** $L=24 ft$ and **Width** $W=6 ft$.

## Example

The **rectangular** pool has a **perimeter** of 5656 meters. The **length** of the pool is given as 1616 meters.

(a) Find the **width** of the pool.

(b) Find the **area** of the pool.

Given information:

The **perimeter** of the pool is $P=5656 m$

The **length** of the pool is $L = 1616 m$

Part a:

We know the **formula** for the **perimeter** of the rectangle is the sum of all **sides** and its formula is given as:

\[P = 2L + 2W \]

Inserting the value of **perimeter** and the **length:**

\[56 = 2(16) + 2W \]

Simply and solving for **Width** $W$:

\[ 56 = 32 + 2W \]

\[ 56 – 32= 2W \]

\[ \dfrac{24}{2} = W \]

**Width** $W$ comes out to be:

\[ W = 12\]

Part b:

The formula for the **Area** of a rectangle is given:

\[A=L \times W\]

Inserting the **values** $L=16$ and $W=12$ in the **formula:**

\[A = 16 \times 12\]

The **area** comes out to be:

\[ A = 192 m^2 \]