# A sheet of paper is 0.1 mm thick. When it's folded in half once, the folded sheet is 0.2 mm thick. Folding it in half again produces a stack 0.4 mm thick. Define the function g so that g(n) is the...

A sheet of paper is 0.1 mm thick. When it's folded in half once, the folded sheet is 0.2 mm thick. Folding it in half again produces a stack 0.4 mm thick. Define the function *g* so that *g(n)* is the thickness when folded n times.

a) Is *g* discrete?

b) Find a formula for *g(n)*

c) How thick is the stack after 8 folds?

d) How many folds are required to obtain a stack that reaches at least the moon, which is 380,000 km from earth?

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(1) `g ` is discrete sine the inputs are natural numbers (1,2,3,...)

(2) `g(n)=.1(2)^(n-1) ` for `n in NN `

This is a geometric sequence with initial value 0.1 and constant ratio `r=2 ` .

(3) After 8 folds the stack is `.1(2)^7=12.8"mm" ` **

(4) If it were possible we can set up the equation:

`.1(2)^(n-1)=3.8"x"10^(11) ` (converting 380000km to mm)

`(2)^(n-1)=3.8"x"10^12 `

`(n-1)ln(2)=ln(3.8"x"10^12) `

`n-1~~41.789 `

(If you do not know logarithms, you could use guess and check to find that n=42 gives a height of approximately 220000km while n=43 gives a height of approximately 440000km.)

So n is approximately 43 folds.**

** There is a limiting factor given the initial thickness of the paper regardless of the size of the paper. This was discovered and proven by a high school student in 2001. See the link.

Also 51 folds will get you to the sun -- if it were physically possible.

a) *g *is not discrete for it is not possible to have negative *g *realistically.

b) g(n) = 2^n

c) g(8) = 2^8

= 256 (mm)

d) 380,000km = 38,000,000,000

g(x) = 380,000,000,000

log2 380,000,000,000 to get answer

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