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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(n) [/tex] that determines the thickness of the folded paper (in mm) in terms of the number of folds made, [tex] n [/tex].

B. The function [tex] g [/tex] has an inverse. The inverse 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 formula for [tex] g^{-1} (t) [/tex].

C. Use your function from part (B) to determine how many times you must fold a piece of paper to make the folded paper have a thickness equal to the distance from the Earth to the Moon. (Assume the distance from the Earth to the Moon is 384,472,300,000 mm).

Answer :

Answer:

(a)[tex]g(n)=0.05\cdot 2^n[/tex]

(b)[tex]g^{-1}(n)=\log_{2}20g(n)[/tex]

(c)43 times

Step-by-step explanation:

Part A

The paper's thickness = 0.05mm

When the paper is folded, its width doubles (increases by 100%).

The thickness of the paper grows exponentially and can be modeled by the function:

[tex]g(n)=0.05(1+100\%)^n\\\\g(n)=0.05\cdot 2^n[/tex]

Part B

[tex]g(n)=0.05\cdot 2^n\\2^n=\dfrac{g(n)}{0.05}\\ 2^n=20g(n)\\$Changing to logarithm form, we have:\\\log_{2}20g(n)=n\\$Therefore:\\g^{-1}(n)=\log_{2}20g(n)[/tex]

Part C

If the thickness of the paper, g(n)=384,472,300,000 mm

Then:

[tex]g^{-1}(n)=\log_{2}20g(n)\\g^{-1}(n)=\log_{2}20\times 384,472,300,000\\=\dfrac{\log 20\times 384,472,300,000}{\log 2} \\g^{-1}(n)=42.8 \approx 43\\n=43[/tex]

You must fold the paper 43 times to make the folded paper have a thickness that is the same as the distance from the earth to the moon.

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Rewritten by : Barada

Final answer:

By using the inverse function, we can determine that you would need to fold the paper approximately 43 times to achieve a thickness equal to the distance from the earth to the moon.

Explanation:

A. Write a function g that determines the thickness of the folded paper (in mm) in terms of the number folds made, n.

To determine the thickness of the folded paper, we can use the formula for exponential growth, which states that the thickness after each fold will be twice the thickness before the fold. Therefore, the function g(n) = 0.05 * 2^n.

B. The function g has an inverse. The function g determines the number of folds needed to give the folded paper a thickness of t mm. Write a function formula for g-1.

To find the number of folds needed for a given thickness, we can use the formula for logarithmic growth, which allows us to solve for the exponent of the exponential function. Therefore, the inverse function g-1(t) = log2(t/0.05).

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.

Using the provided distance from the earth to the moon (384,472,300,000 mm), we can substitute this value into the inverse function g-1(t) = log2(t/0.05) to find the number of folds required. Plugging in the value gives us g-1(384,472,300,000) = log2(384,472,300,000/0.05) ≈ 42.3687. Therefore, you would need to fold the paper approximately 43 times to achieve a thickness equal to the distance from the earth to the moon.

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