The length of an algebra 2 textbook is 2 times the height. the sum of the length, width, and height of the box is 10 centimeters. a. write the expressions for the dimensions of the book. b. write a polynomial function for the volume of the book in factored form. c. find the maximum volume of the book.

As per the given statement:

The length of an Algebra 2 textbook is 2 times the height.

Let height be x then;

⇒   ......[1]; where l is the length.

Also, the sum of the length, width and height of the box is 10 cm.

⇒ where w is the width.      .,.....[2]

Substitute equation [1] in [2] we get;

oe

or

                          ......[3]

(a)

The dimensions of the box is :-

(b)

Volume of the book is given by:

where V is the volume.

Substitute equation [1] and [3] in above formula;

The polynomial function for the volume of the book in the factored form:

(c)

To find the maximum volume of the book;

we would find the derivative of volume with respect to x i.e,

V(x) =        ......[4]

Now;

Set this derivative equal to 0.

or

By Zero Product Property states that if ab = 0, then

either a = 0 or b = 0, or we have both a and b are 0.

then we have;

and

Then substitute these values in equation [4] to get the values of V(x);

and  

= 32.92(apporx)

So,  V(x) = 32.92 which is maximum for

Therefore, the graph of function V(x) is shown below and we can clearly see that there is a maximum very close to 2.22..