Question

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

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

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.

$x$

$2x$

$x$

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.

$A''(x)=-2$

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.