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. ...?

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

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erinna erinna · Answer 2 · 2019-10-14T09:43:11+02:00

Answer:

Area function : [tex]A(x)=12x-x^2[/tex]

Domain: (0,6)

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

Step-by-step 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).

[tex]24 = 2( x + \text{longest side})[/tex]

[tex]12 = x + \text{longest side}[/tex]

[tex]12 - x = \text{longest side}[/tex]

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

Area of a rectangle is

[tex]A=length \times width[/tex]

Area function is

[tex]A(x)=x(12-x)[/tex]

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

[tex]A(x)>0[/tex]

[tex]x(12-x)>0[/tex]

It means,

[tex]x>0[/tex]

[tex]12-x>0\Rightarrow 12>x[/tex]

One side is less that the other side.

[tex]x<12-x[/tex]

[tex]2x<12[/tex]

[tex]x<6[/tex]

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

The simplified form of the area function is

[tex]A(x)=12x-x^2[/tex]

Differentiate with respect to x.

[tex]A'(x)=12-2x[/tex]

[tex]A'(x)=0[/tex]

[tex]12-2x=0[/tex]

[tex]x=6[/tex]

Differentiate A'(x) with respect to x.

[tex]A''(x)=-2<0[/tex]

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

[tex](12-x)=12-6=6[/tex]

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