College

A standard piece of paper is 0.05 mm thick. Let's imagine taking a piece of paper and folding it in half multiple times. We'll assume we can make "perfect folds," where each fold makes the folded paper exactly twice as thick as before, and we can make as many folds as we want.

a. Write a function [tex]\( g \)[/tex] that determines the thickness of the folded paper (in mm) in terms of the number of folds made, [tex]\( n \)[/tex]. (Notice that [tex]\( g(0) = 0.05 \)[/tex].)

[tex]\[ g(n) = \][/tex]

[tex]\(\square\)[/tex]

b. The function [tex]\( g \)[/tex] has an inverse. The function [tex]\( g^{-1} \)[/tex] determines the number of folds needed to give the folded paper a thickness of [tex]\( t \)[/tex] mm. Write a function formula for [tex]\( g^{-1} \)[/tex].

[tex]\[ g^{-1}(t) = \][/tex]

[tex]\(\square\)[/tex]

c. Use your function in part (b) to determine how many times you must fold a piece of paper to make the folded paper have a thickness that is the same as the distance from the Earth to the Moon. (Assume the distance from the Earth to the Moon is [tex]\( 384,472,300,000 \)[/tex] mm).

[tex]\(\square\)[/tex] folds

Answer :

Sure! Let's tackle each part of the question step-by-step.

### Part (a): Write a function [tex]\( g(n) \)[/tex]

We want to determine the thickness of the paper after folding it [tex]\( n \)[/tex] times. Each fold doubles the thickness. If the initial thickness of the paper is 0.05 mm, after 1 fold the thickness is [tex]\( 2 \times 0.05 \)[/tex] mm. After 2 folds, it is [tex]\( 2 \times 2 \times 0.05 \)[/tex] mm, and so on.

To express this relationship as a function, we use:
[tex]\[ g(n) = 0.05 \times 2^n \][/tex]
where [tex]\( g(n) \)[/tex] is the thickness in millimeters after [tex]\( n \)[/tex] folds.

### Part (b): Write the inverse function [tex]\( g^{-1}(t) \)[/tex]

The inverse function [tex]\( g^{-1}(t) \)[/tex] helps us find the number of folds needed to reach a certain thickness [tex]\( t \)[/tex]. We start from the equation:
[tex]\[ t = 0.05 \times 2^n \][/tex]

To solve for [tex]\( n \)[/tex], we need to isolate it:
1. Divide both sides by 0.05:
[tex]\[ \frac{t}{0.05} = 2^n \][/tex]
2. Use the logarithm (base 2) to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \log_2\left(\frac{t}{0.05}\right) \][/tex]

So, the inverse function is:
[tex]\[ g^{-1}(t) = \log_2\left(\frac{t}{0.05}\right) \][/tex]

### Part (c): Determine the number of folds needed for a thickness equal to the distance from Earth to the Moon

The distance from Earth to the Moon is given as 384,472,300,000 mm. We apply the inverse function [tex]\( g^{-1}(t) \)[/tex] from Part (b) to find the number of folds needed:
[tex]\[ n = \log_2\left(\frac{384472300000}{0.05}\right) \][/tex]

By calculating the above expression, we find that:
[tex]\[ n \approx 42.81 \][/tex]

Since we cannot have a fraction of a fold, we'll need to fold the paper approximately 43 times for the thickness to reach or exceed the distance from the Earth to the Moon.

So, the number of folds needed is about 43 folds.