Chapter 4.4, Problem 5MRPS

Solution

a.

Copy and complete the table.

The solution is shown in the table.

Given information:

A piece of paper is folded in half; the paper is divided into regions each of which has half the area of the paper.

  $\begin{array}{cccccc}\text{Fold}\text{number}& 0& 1& 2& 3& 4\\ \text{Number}\text{of}\text{regions}& 1& 2& ?& ?& ?\\ \text{Fractional}\text{area}\text{of}\text{each}\text{region}& 1& \frac{1}{2}& ?& ?& ?\end{array}$

Concept Used:

Use the formula: Fold number / Number of regions

Calculation:

As per the given problem.

Note that the total number of regions doubles after each fold. The area of each resulting region is half the previous region area.

  $\begin{array}{ccc}\text{Fold}\text{number}& \text{Number}\text{of}\text{regions}& \text{Fractional}\text{area}\text{of}\text{ecah}\text{region}\\ 0& 1& 1\\ 1& 2& \frac{1}{2}\\ 2& 4& \frac{1}{4}\\ 3& 8& \frac{1}{8}\\ 4& 16& \frac{1}{16}\end{array}$

Conclusion:

Hence, the solution is in the table.

b.

Write the function giving the number of regions $R\left(n\right)$ and the fractional area of each region $A\left(n\right)$ after $n$ folds.

Therefore the fractional area decreases exponentially.

Given information:

A piece of paper is folded in half; the paper is divided into regions each of which has half the area of the paper.

  $\begin{array}{cccccc}\text{Fold}\text{number}& 0& 1& 2& 3& 4\\ \text{Number}\text{of}\text{regions}& 1& 2& ?& ?& ?\\ \text{Fractional}\text{area}\text{of}\text{each}\text{region}& 1& \frac{1}{2}& ?& ?& ?\end{array}$

Concept Used:

Use the formula: $R\left(n\right)\times A\left(n\right)$ = Area of the paper

Calculation:

As per the given problem

Note that the number of regions $R\left(n\right)=2^n$ , the number of regions increase exponentially with the increase in the number of folds.

Note that $R\left(n\right)\times A\left(n\right)$ = Area of the paper $=1$ from the table,

Therefore,

  $A\left(n\right)=\frac{1}{R\left(n\right)}=2^{-n}$.

Therefore the fractional area decreases exponentially.

Conclusion:

Therefore the fractional area decreases exponentially.