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------------------------------------------------ 5. An unfolded piece of paper is 0.05 mm thick.

a. Complete the table with the thickness of the piece of paper [tex] \( T(n) \) [/tex] after it is folded in half [tex] \( n \) [/tex] times.

\[
\begin{tabular}{|c|c|}
\hline
[tex] $n$ [/tex] & [tex] $T(n)$ [/tex] \\
\hline
0 & 0.05 \\
\hline
1 & 0.1 \\
\hline
2 & 0.2 \\
\hline
3 & 0.4 \\
\hline
\end{tabular}
\]

b. What is a reasonable domain for the function [tex] \( T \) [/tex]? Explain how you know.

Answer :

a. Updated Table will be

n T(n) (mm)

0 0.05

1 0.1

2 0.2

3 0.4

b. The domain can be written as: n ∈ {0, 1, 2, …, n_max},

a. The thickness of the paper after being folded n times doubles with each fold. The formula for the thickness is: T(n) = 0.05 × 2ⁿ (in mm).

Using this formula:

T(0) = 0.05 mm

T(1) = 0.05 × 2 = 0.1 mm

T(2) = 0.05 × 4 = 0.2 mm

T(3) = 0.05 × 8 = 0.4 mm

b.T(n) models the thickness of the paper as it is folded. A reasonable domain is the set of non-negative integers (n ≥ 0) because:

The number of folds, n, cannot be negative.

In real life, there is a practical limit to how many times a piece of paper can be folded. For a standard sheet of paper, this limit is usually 7-8 folds due to physical constraints.

Complete question

a. Complete the table with the thickness of the piece of paper after it is folded in half times.

n T(n) (mm)

0

1

2

3

b. What is a reasonable domain for the function?

To solve this problem, let's first understand what is being asked.

Part A: Completing the Table

The table shows the thickness of the piece of paper after it is folded in half multiple times. Each time you fold the paper in half, the thickness doubles.

Let's calculate the thickness of the paper after it is folded n times:

  1. Initial Thickness[tex]\,(n = 0)[/tex]:

    • The paper starts at 0.05 mm thick.
    • [tex]T(0) = 0.05[/tex]

  2. First Fold[tex]\,(n = 1)[/tex]:

    • Fold the paper once, doubling its thickness:
    • [tex]T(1) = 0.05 \times 2 = 0.1[/tex]

  3. Second Fold[tex]\,(n = 2)[/tex]:

    • Fold again, doubling the thickness:
    • [tex]T(2) = 0.1 \times 2 = 0.2[/tex]

  4. Third Fold[tex]\,(n = 3)[/tex]:

    • Fold again, doubling the thickness:
    • [tex]T(3) = 0.2 \times 2 = 0.4[/tex]

From this pattern, we can observe that the n-th fold gives us a thickness [tex]T(n) = 0.05 \times 2^n[/tex].

Part B: Determining a Reasonable Domain

For the function [tex]T(n)[/tex], the domain represents the number of times you fold the paper. Since [tex]n[/tex] represents the number of folds, it is:

  • A non-negative integer (you can't fold a paper a negative number of times).
  • Technically limited by physical constraints of folding paper, as it becomes increasingly difficult to fold beyond 7-8 times. However, theoretically, [tex]n[/tex] can be any whole number starting from 0.

Thus, a reasonable domain for this function is the set of non-negative integers, usually represented as [tex]n \geq 0[/tex].

Missing Table from question added below:

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