Question

A wire 24inches long is to be cut into four pieces to form a rectangle whose shortest side has a length of x:

Determine the domain of the function and use a graphing utility to graph the function over that domain
Use the graph of the function to approximate the maximum area of the rectangle. Make a conjecture about the dimensions that yield a maximum area. ...?

Answer 1

If the function is the Area, then Area = length*width = x*(12-x) = 12x - x^2Domain is x>0, since you can't have a rectangle with negative length.

0 < x<6, If x is 7 then width would be 5, but x must be shorter

Answer 2

Area function :
`A(x)=12x-x^2`

Domain: (0,6)

The area of rectangle is maximum at x=6. The area of a rectangle is maximum if it is a square.

Explanation:

It is given that the length of wire is 24 inches. It is to be cut into four pieces to form a rectangle.

Let x be the length of shortest side.

Perimeter of a rectangle is

Perimeter = 2( Shortest side + longest side).

`24 = 2( x + \text{longest side})`

`12 = x + \text{longest side}`

`12 - x = \text{longest side}`

So, length of longest side is (12-x) inches.

Area of a rectangle is

`A=length * width`

Area function is

`A(x)=x(12-x)`

The area of rectangle and dimensions of a rectangle can not be a negative.

`A(x)>0`

`x(12-x)>0`

It means,

`x>0`

`12-x>0\Rightarrow 12>x`

One side is less that the other side.

`x<12-x`

`2x<12`

`x<6`

It means the domain of the function is (0,6).

The simplified form of the area function is

`A(x)=12x-x^2`

Differentiate with respect to x.

`A'(x)=12-2x`

`A'(x)=0`

`12-2x=0`

`x=6`

Differentiate A'(x) with respect to x.

`A''(x)=-2<0`

Therefore the area of rectangle is maximum at x=6.

`(12-x)=12-6=6`

It means the area of a rectangle is maximum if it is a square.