# Question #ec7ee

Mar 8, 2017

The dimensions are $12 x \text{ by " 8x " by } 4 x$ inches.

#### Explanation:

Let's look at the actual dimensions of the carton first.

The length of the carton will be the width of three dictionaries placed side by side. $4 + 4 + 4 = 12$ inches

The width of the carton will be the length of one book: $8$ inches

The height will be the thickness of four books on top of each other:
$1 + 1 + 1 + 1 = 4$ inches

So the carton is $12 \text{ by " 8 " by } 4$ inches.

As a polynomial we have to write the dimensions of each book in terms of $x$.
Let's call the thickness $x$
The width of each book is therefore $4 x$
and the length is $8 x$

The dimensions of the box are therefore:

Length: $4 x + 4 x + 4 x = 12 x$
Width: $8 x$
Height: $x + x + x + x = 4 x$

We can only write a single polynomial if we know what we are doing with the separate dimensions.

If it is the volume of the carton, it would be:

$V = l \times b \times h = 12 x \times 8 x \times 4 x = 384 {x}^{3}$

If it is the total surface area it would be:

$T S A = 2 \left(l b + l h + b h\right) = 2 \left(12 x \times 8 x + 12 x \times 4 x + 8 x \times 4 x\right)$

$= 2 \left(96 {x}^{2} + 48 {x}^{2} + 32 {x}^{2}\right)$

$= 2 \left(176 {x}^{2}\right) = 352 {x}^{2}$