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 = lengthwidth = 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 Link

erinna erinna · Answer 2 · 2019-10-14T03:43:11-04:00

Answer:

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.

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).

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

\(12 = x + \text{longest side}\)

\(12 - x = \text{longest side}\)

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

Area of a rectangle is

\(A=length \times width\)

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.