# The length of a rectangular sign is 5 times its width. If the sign's perimeter is 24 inches, what...

## Question:

The length of a rectangular sign is 5 times its width. If the sign's perimeter is 24 inches, what is the sign's area?

## Area, Perimeter, and Properties of a Rectangular Shape:

Rectangles - They are those quadrilaterals whose opposite sides are parallel and equal in size and there is a right angle at each corner.

There are four corners in a rectangle.

A square is also a quadrilateral.

It has two diagonals and diagonal length is calculated bu using the formula written below-

Using the Pythagorean theorem-

{eq}\displaystyle d^{2} = l^{2}+w^{2} {/eq}

here d is the diagonal length of the rectangle.

l and w are the length and width of the rectangle.

#### Area of the Rectangle

{eq}\displaystyle A = l \times w {/eq}

here A is the area of the rectangle

#### Perimeter of the Rectangle

{eq}\displaystyle P = 2(l+w) {/eq}

here P is the Perimeter of the Rectangle

Given that the length of a rectangular sign is 5 times its width.

{eq}\displaystyle l = 5+w --------(1) {/eq}

Also given that the rectangular sign's perimeter is {eq}24~ inches {/eq}

{eq}\displaystyle P = 24 ~inches {/eq}

{eq}\displaystyle 2(l+w) = 24 {/eq}

{eq}\displaystyle l+w = 12 {/eq}

Now put the value of l from the equation(1)-

{eq}\displaystyle 5+w+w = 12 {/eq}

{eq}\displaystyle 2w = 7 {/eq}

{eq}\displaystyle w = \frac{7}{2} = ~3.5 ~inches {/eq}

from the equation(1)-

{eq}\displaystyle l = 5+3.5 = 8.5 ~inches {/eq}

So the area of the sign-

{eq}\displaystyle A = 3.5 \times 8.5 = 29.75 ~inches {/eq}

So the area of the rectangular sign is {eq}29.75 ~inches. {/eq}