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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 384,472,300,000 mm.)

Answer :

The paper must be folded 43 times to get the thickness same as the distance from the earth to the moon.

Note that the function g is defined as [tex]g(n)=0.052^n.[/tex]

Let g(n) be the thickness that is g(n)=t.

Thus,

[tex]t & =0.05\left(2^n\right) \\\frac{t}{0.05} & =2^n \\\frac{100 t}{5} & =2^n \\20 t & =2^n\end{aligned}[/tex]

Take logarithms on both sides, thus

[tex]$$\begin{aligned}\log (20 t) & =\log \left(2^n\right) \\\log (20 t) & =n \log (2) \\n & =\frac{\log (20 t)}{\log (2)} \\n & =\log _2 20 t\end{aligned}$$[/tex]

Thus, [tex]g^{-1}=\log _2 20t[/tex] is the required inverse function.

c) Given that the thickness of the paper is same as the distance from the earth to the moon that is t=384,472,300,000 mm.

To determine the corresponding value of n.

Note that the inverse function is defined as [tex]g^{-1}(t)=\log _2 20t[/tex]...

Substitute 384,472,300,000 in equation (1), thus

[tex]$$\begin{aligned}& g^{-1}(384,472,300,000)=\log _2(20 \times 384,472,300,000) \\& g^{-1}(384,472,300,000)=\log _2(20 \times 384,472,300,000) \\& g^{-1}(384,472,300,000)=42.806\end{aligned}$$[/tex]

Since, [tex]g^{-1}[/tex] determines the number of folds of the paper, the value must be the whole number.

Thus, [tex]g^{-1}(384,472,300,000)[/tex] is approximately equal to 43 .

Thus, the paper must be folded 43 times to get the thickness same as the distance from the earth to the moon.

The complete questions should be:

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

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 384,472,300,000 mm. folds

For more questions on inverse function

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