Chapter 7.2, Problem 41PPS

(a)

To determine

To find: the number of count of the sheets in the stack after the first cut and number of pieces after the first, second, and third cut.

Answer to Problem 41PPS

There are $2,4,8$ , and $16$ pieces of sheet after the first, second, third, and fourth cut respectively.

Explanation of Solution

Given information:

A large sheet of paper is cut in half, and one of the resulting pieces is placed on top of the other. Then the pieces in the stack are cut in half and placed on top of each other.

When cut the sheet first time then, the sheet divides into two parts. That means 2 pieces after first cut.

There are two pieces and each piece has two halves. Then, the total pieces are $2·2=4$ .

There are four pieces and each piece has two halves. Then, the total pieces are $4·2=8$ .

There are eight pieces and each piece has two halves. Then, the total pieces are $8·2=8$ .

Therefore, there are $2,4,8$ , and $16$ pieces of sheet after the first, second, third, and fourth cut respectively.

(b)

To determine

To find: the table for the process of first, second, and third cut of sheet.

Answer to Problem 41PPS

The required table is shown in table 1.

Explanation of Solution

Given information:

A large sheet of paper is cut in half, and one of the resulting pieces is placed on top of the other. Then the pieces in the stack are cut in half and placed on top of each other.

As it has found that there are $2,4,8$ , and $16$ pieces of sheet after the first, second, third, and fourth cut respectively.

Construct the table for the cutting process.

CUTS	PIECES
1	2
2	4
3	8

Table 1

Therefore, the required table is shown in table 1.

(c)

To determine

To find: the function for the given process.

Answer to Problem 41PPS

The required function is $y=2^x$ .

Explanation of Solution

Given information:

A large sheet of paper is cut in half, and one of the resulting pieces is placed on top of the other. Then the pieces in the stack are cut in half and placed on top of each other.

Consider the table from b part.

CUTS	PIECES
1	2
2	4
3	8

Table 1

Suppose $y$ represents the number of pieces after each cut and $x$ represents the number of cuts.

So, the function can be written as $y=2^x$ .

Therefore, the required function is $y=2^x$ .

(d)

To determine

To find: the function for the given process and thickness of the paper.

Answer to Problem 41PPS

The required function is $y=0.003\left(2^x\right)$ .

Explanation of Solution

Given information:

A large sheet of paper is cut in half, and one of the resulting pieces is placed on top of the other. Then the pieces in the stack are cut in half and placed on top of each other and the thickness of the ordinary paper is $0.003\text{inch}\text{.}$ .

Consider the Function from c part.

  $y=2^x$

As the thickness of ordinary paper is about $0.003\text{inch}\text{.}$ , then the function can be written as,

  $y=0.003\left(2^x\right)$

Therefore, the required function is $y=0.003\left(2^x\right)$ .

(e)

To determine

the thickness of the paper after $30$ cuts.

Answer to Problem 41PPS

The thickness of the stack of paper after $30$ cuts is about $3,221,225.47\text{inches}$ .

Explanation of Solution

Given information:

A large sheet of paper is cut in half, and one of the resulting pieces is placed on top of the other. Then the pieces in the stack are cut in half and placed on top of each other and the thickness of the ordinary paper is $0.003\text{inch}\text{.}$ .

Consider the Function from c part.

  $y=0.003\left(2^x\right)$

Substitute $30$ for $x$ in the function $y=0.003\left(2^x\right)$ to find the number of pieces.

  $\begin{array}{c}y=0.003\left(2^{30}\right)\\ =3,221,225.47\end{array}$

Therefore, the thickness of the stack of paper after $30$ cuts is about $3,221,225.47\text{inches}$ .