Paper moon problem

First fold: 0.00000014

Second fold: 0.00000028

....

Forty second fold: 307,863

Forty third fold: 615,726

So we do on the calculator: Ans = Ans * 2 (starting with 0.00000007 km) counting the number of operations we make a long the way. When Ans reaches >384,400 we stop counting.

Therefore, the number of operations needed to exceed 384,400 is 43 .

Each folding of the paper doubles the thickness of the paper. That is, if we fold the paper $n$ times, the paper's thickness is $2^n$ times its original thickness. As a result, $7\times10^{-5} \times 2^n = 3.844\times 10^8$ $2^n= \frac{3.844\times 10^8}{7\times10^{-5}} = 5.491 \times 10^{12}$ $n = \log_2(5.491 \times 10^{12}) = 42.32$ Therefore, the answer is $\boxed{43}$ , because 42 folds still falls short so we need an extra fold.