Chapter 11.1, Problem 5E

(a)

To determine

To draw: a diagram that represents the box.

Answer to Problem 5E

  

Explanation of Solution

Given:

Square piece of material, 24 centimeters on a side

Calculation:

Let the sides of the box be $x$ centimeters tall. Then, the length of the box is the original $24$ inches, minus $xcentimeters$ on either side, therefore the length (and width} of the box is $24-2x=2\left(12-x\right)$ .

  

Conclusion:

Thus, the requred diagram is drawn.

(b)

To determine

To verify:the volume $V$ of the box is $V=4x{\left(12-x\right)}^2$ .

Explanation of Solution

Given:

  $V=4x{\left(12-x\right)}^2$

In Part A, the length and width of the box are $2\left(12-x\right)$ , and the height is $x$ . so the volume is given by:

  $V=l\cdot w\cdot h=2\left(12-x\right)\cdot 2\left(12-x\right)\cdot x=4x{\left(12-x\right)}^2$

Conclusion:

Therefore, the volume $V$ of the box is $V=4x{\left(12-x\right)}^2$ .

(c)

To determine

To complete: the table,observe the behavior and find the limit.

Answer to Problem 5E

The value of the limit is $\underset{x\to 4}{\lim }V=1024$

Explanation of Solution

Given:

  $\underset{x\to 4}{\lim }V$

Calculation:

Part C

$x$	3	3.5	4	4.1	4.5	5
$V$	972	1011.5	1024	1023.52	1012.5	980

from the table above. thefunction appears to take values close to $1024asxapproaches4$ from both the left and right,thus, $\underset{x\to 4}{\lim }V=1024$

Conclusion:

The value of the limit is $\underset{x\to 4}{\lim }V=1024$

(d)

To determine

To graph: the volume function and verify the volume is maximum when $x=4$ .

Answer to Problem 5E

  

Explanation of Solution

Calculation:

Graphing this function, it takes its highest value when $x=4$

  

Conclusion:

Therefore, the volume is maximum when $x=4$ .