# A rectangular area is to be enclosed by a wall on one side and fencing on the other three sides....

## Question:

A rectangular area is to be enclosed by a wall on one side and fencing on the other three sides. If 18 meters of fencing are used, what is the maximum area that can be enclosed?

## The Perimeter and Area of a Rectangle:

A rectangle is a two-dimensional figure that has two pairs of parallel sides. The parallel sides of a rectangle are congruent, and each of the interior angles is a right angle. The area of a rectangle is the amount of space that is covered by a rectangular figure. We determine this area by multiplying the length of the rectangular figure by its width.

To determine the maximum area that can be enclosed, we will make use of the perimeter formula and the area formula.

The perimeter of a rectangle is computed as:

• {eq}P = 2l + 2w {/eq}

And the area is calculated as:

• {eq}A = lw {/eq}

One of the sides is enclosed by a wall. Therefore, the length of the three sides enclosed by a fence will be given by:

• {eq}P = l + 2w {/eq}

If the fence is 18m, then:

• {eq}18 = l + 2w {/eq}
• {eq}l = 18 - 2w {/eq}............................................................................................(i)

Substituting equation (i) into the area formula:

• {eq}A = (18 - 2w)w = 18w - 2w^2 {/eq}

Maximizing the area:

• {eq}\dfrac{d A}{d w} = 18 - 4w = 0 {/eq}
• {eq}18 - 4w = 0 {/eq}

Solving for w:

• {eq}w = \dfrac{18}{4} = 4.5\, m {/eq}

Therefore, the maximum area is equal to:

• {eq}A = 18(4.5) - 2(4.5)^2 = \boxed{40.5\, m^2} {/eq}