Chapter 11.4, Problem 50PPS

(a)

To determine

List fractions represents by the pieces.

Answer to Problem 50PPS

The lists of the fractions are $\frac{1}{2};\frac{1}{4};\frac{1}{8};\frac{1}{16};\frac{1}{32};\mathrm{.........}$

Explanation of Solution

Given:

In this problem, you will use a square of paper that is at least 8 inches on a side.

Let the square be one unit. Cut away one half of the square. Call this piece Term 1. Next cut away one half of the remaining sheet of paper. Call this piece Term 2. Continue cutting the remaining paper in half and labeling the pieces with a term number as long as possible. List fractions represents by the pieces.

Calculation:

  $\frac{1}{2};\frac{1}{4};\frac{1}{8};\frac{1}{16};\frac{1}{32};\mathrm{.........}$

Thus, the lists of the fractions are $\frac{1}{2};\frac{1}{4};\frac{1}{8};\frac{1}{16};\frac{1}{32};\mathrm{.........}$

(b)

To determine

Find the sum of the series.

Answer to Problem 50PPS

The sum of the series is 1.

Explanation of Solution

Given:

In this problem, you will use a square of paper that is at least 8 inches on a side.

Let the square be one unit. Cut away one half of the square. Call this piece Term 1. Next cut away one half of the remaining sheet of paper. Call this piece Term 2. Continue cutting the remaining paper in half and labeling the pieces with a term number as long as possible. List fractions represents by the pieces.

If you could cut the squares indefinitely, you would have an infinite series. Find the sum of the series.

Concept Used:

Initial is $a_1=\frac{1}{2}$

  $a_1=\frac{1}{2};r=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$

Calculation:

Now use the formula for sum:

  $\begin{array}{l}{a}_1=\frac{1}{2};r=\frac{1}{2};\text{Find}\text{the}\text{Sum}\\ \text{S = }\frac{a_1}{1-r}=\frac{\frac{1}{2}}{1-\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{1}{2}}=1\end{array}$

The sum of the series is 1.

Thus, the sum of the series is 1.

(c)

To determine

Find how the sum of the series relates to the original square of Paper.

Answer to Problem 50PPS

The original square has are 1 unitand the area of all pieces cannot exceed 1.

Explanation of Solution

Given:

How does the sum of the series relate to the original square of Paper?

Concept Used:

The original square has are 1 unit and the area of all pieces cannot exceed 1.

Thus, the original square has are 1 unitand the area of all pieces cannot exceed 1.