# If you have 300 meters of fencing and want to enclose a rectangular area up against a long,...

## Question:

If you have 300 meters of fencing and want to enclose a rectangular area up against a long, straight wall, what is the largest area you can enclose?

## The Perimeter and Area of a Rectangle:

A rectangle is a plane figure that has four sides and four right angles. The opposite sides in a rectangle are equal, but the adjacent sides are not. The perimeter of a square is twice the sum of its length and width, while the area is the length multiplied the width.

The perimeter of a rectangle is calculated as:

• {eq}P = 2(l + w) {/eq}

And the area is calculated as:

• {eq}A = l\times w {/eq}

If there are 300 meters of fencing, and we want to enclose a rectangular area up against a rectangular wall, then the perimeter of this area will be given by:

• {eq}300 = 2(l + w) - l {/eq}
• {eq}300 = l + 2w {/eq}

Solving for l, we get:

• {eq}l = 300 - 2w {/eq}

The area of the enclosed field will be equal to:

• {eq}A = (300 - 2w)w = 300w - 2w^2 {/eq}

The area of the field assumes a rectangular equation. Therefore, the maximum area will occur at the point where:

• {eq}w = \dfrac{-b}{2a} {/eq}

{eq}a = -2, \quad b = 300 {/eq}. Therefore:

• {eq}w = \dfrac{-300}{2(-2)} = 75\; \rm m {/eq}

Therefore, the largest area that we can enclose using 300 meters of fencing is:

• {eq}A = (300 - 2\times 75)75 {/eq}
• {eq}\boxed{\color{blue}{A = 11,250\; \rm m}} {/eq}