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A piece of paper is 0.1 mm thick. When folded, the paper is twice as thick. Assume that you could...

Question:

A piece of paper is {eq}\rm 0.1\ mm {/eq} thick. When folded, the paper is twice as thick. Assume that you could fold the paper as many times as you want. How many folds would be required for the paper to be taller than Mount Everest {eq}\rm (8850 \ m) {/eq}?

Mount Everest

Mount Everest is the highest mountain on the surface of the Earth above sea level. During the climbing season, which occurs between April and May, it would take hikers about two months to climb Mount Everest.

Answer and Explanation:

Note that in every fold, the thickness of the paper doubles. The resulting thickness after the first fold is {eq}2\times 0.1\,\rm mm=0.2\,\rm mm {/eq}, after the second fold, its {eq}2\times.02\,\rm m=0.4\,\rm mm {/eq}, after the the third fold, its {eq}2\times.04\,\rm m=0.8\,\rm mm {/eq}, and so on. Therefore, the thickness {eq}T_{n} {/eq} after {eq}n {/eq} folds is given by

$$T_{n}=(0.1\,\rm mm)2^n $$

By rearranging the above equation, we have

$$\begin{align} 2^n &=\dfrac{T_{n}}{0.1\,\rm mm}\\[0.2cm] \ln \mid 2^n \mid &=\ln \left| \dfrac{T_{n}}{0.1\,\rm mm} \right|\\[0.2cm] n &=\dfrac{\ln \left| \tfrac{T_{n}}{0.1\,\rm mm} \right|}{\ln \mid 2 \mid} \end{align} $$

Then, we convert the height of Mount Everest from m to mm.

$$\begin{align} 8850\, \rm m &=8850\, \rm m \left(\dfrac{1000\,\rm mm}{1\rm m}\right)\\[0.2cm] 8850\, \rm m &=8,850,000\,\rm m \end{align} $$

Therefore, the number of folds needed for the paper to be taller than Mount Everest would be

$$\begin{align} n &>\dfrac{\ln \left| \tfrac{8,850,000\,\rm m} {0.1\,\rm mm}\right|}{\ln \mid 2 \mid}\\[0.2cm] n &>\dfrac{18.30}{0.69}\\[0.2cm] n&> 26\,\rm folds \end{align} $$

Therefore, the paper should be folded at least 27 times for the resulting thickness to be taller than Mount Everest.


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