$\"\"$

(a)

The completed table :

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

Height, $\"\"$	Length and Width, $\"\"$	Volume, $\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$

Observe the above table :

The maximum volume occurs at $\"\"$ , i.e, $\"\"$ .

$\"\"$

(b)

From the table :

Volume of the box :

$\"\"$

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

(c)

Volume $\"\"$ .

Derivative on each side with respect to $\"\"$ .

$\"\"$

Find the crirical numbers by equating derivative to zero.

$\"\"$

$\"\"$ and $\"\"$

$\"\"$ and $\"\"$ .

Thus, the critical numbers are $\"\"$ and $\"\"$ .

$\"\"$

$\"\"$ .

Derivative on each side with respect to $\"\"$ .

$\"\"$

$\"\"$ .

If $\"\"$ , then $\"\"$ .

By second derivative test, $\"\"$ is a maximum value.

Substitute $\"\"$ in $\"\"$ .

$\"\"$ .

Thus, the maximum volume is $\"\"$ .

If $\"\"$ , then $\"\"$ .

By the second derivative test, $\"\"$ is a minimum value.

$\"\"$

The volume function is $\"\"$ .

Draw a coordinate plane.

Graph the function $\"\"$ .

Graph :

$\"\"$

Observe the graph :

The maximum volume occurs at $\"\"$ and the maximum volume is $\"\"$ .

$\"\"$

(a)

The completed table :

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

Height, $\"\"$	Length and Width, $\"\"$	Volume, $\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$
$\"\"$	$\"\"$	$\"\"$

The maximum volume occurs at $\"\"$ , i.e, $\"\"$ .

(b)

Volume $\"\"$ .

(c)

The critical numbers are $\"\"$ and $\"\"$ .

The maximum volume is $\"\"$ .

(d)

Graph of the function $\"\"$ :

$\"\"$

The maximum volume occurs at $\"\"$ and the maximum volume is $\"\"$ .