# Math

Characteristics of Quadratic Functions Example

Hello,

I do not understand step#4. Where did the -w come from? Everything else I understand competely.

THanks,

MB

A rectangular field is to be enclosed by 600m of fence.

• What is the maximum area?
• What dimensions will give a maximum area?
• Step by Step...

Step 1:

Let l be the length of the rectangleLet w be the width of the rectangleLet A be the area of the rectangle.

Step 2:

Draw a diagram if possible, and label with appropriate variables

Step 3:

Write down the equations.

The 600m fencing means the perimeter of the rectangle is 600 m. Therefore, the first equation is:

2l + 2w = 600 ---- (1)

You were asked to maximize the area of the rectangle, so the second equation is:

A = l × w --- (2)

Step 4:

Get a quadratic function.

Use equation (1) to solve for or w. Let’s say you solve for l.

l = −w + 300 ---- (3)

## Expert Answers

You need to remember the equation that gives the area of rectangle, such that:

`A = l*w`

l represents the length

w represents the width

Since the problem provides the value of perimeter of rectangle, you may write area, either in terms of length, or in terms of width, using the equation of perimeter, such that:

`600 = 2(l + w) => 300 = l + w => l = 300 - w`

Hence, substituting back 300 - w for l in equation of area yields:

`A(w) = (300 - w)*w`

Opening the brackets yields:

`A(w) = 300w - w^2`

You need to maximize the area of rectangle, hence, you need to differentiate the function `A(w) = 300w - w^2`  with respect to w, such that:

`A'(w) = (300w - w^2)' => A'(w) = 300*w' - (w^2)'`

`A'(w) = 300*1 - 2w => A'(w) = 300 - 2w`

Now, you need to solve for w the equation `A'(w) = 0`  such that:

`300 - 2w = 0 => 2w = 300 => w = 150 m`

Hence, evaluating the dimensions that maximize the area of the fence, yields `w=l = 150 m` .

Approved by eNotes Editorial Team