$\"\"$

(a).

Length of the box $\"\"$ .

Height of the box $\"\"$ .

Width of the box $\"\"$ .

Volume of the box, $\"\"$ .

$\"\"$ .

$\"\"$

$\"\"$ .

The polynomial function is $\"\"$ .

$\"\"$

(b).

Length of the box must be $\"\"$ .

Height of the box must be $\"\"$ .

Width of the box must be $\"\"$ or $\"\"$ .

Therefore, the domain of $\"\"$ is $\"\"$ .

Subustiute the values of $\"\"$ in the obtained polynomial equation to get the values of $\"\"$ .

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Graph :

(1) Draw the coordinate plane.

(2) Plot the point obtained in the above table.

(3) Connect the points with the smooth curve.

$\"\"$

(c).

The company wants the box to have volume of $\"\"$ .

Substiute the value of $\"\"$ in the obtained polynomial equation $\"\"$ .

$\"\"$ .

$\"\"$

(d).

$\"\"$ .

$\"\"$ when $\"\"$ .

$\"\"$ .

Use synthetic substitution to verify $\"\"$ .

$\"\"$

The remainder is $\"\"$ .

So $\"\"$ is a factor of $\"\"$ .

Therefore, $\"\"$ is a solution for $\"\"$ .

$\"\"$

(a).

The polynomial function is $\"\"$ .

(b).

Graph :

$\"\"$

(c).

$\"\"$ .

(d).

$\"\"$ is a solution for $\"\"$ .