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A small plane is flying a banner in the shape of a rectangle. The area of the banner is 144 square feet. The width of the banner is 1/4 the length of the banner. What are the dimensions of the banner?

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:

\[ A = 192 m^2 \]

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