Answer
$\$ 10,737,418.23$,
37 days.
Work Step by Step
Let $a_{n}$ denote the amount set aside on the nth day.
First term: 0.01 (dollars),
common ratio: 2.
This is a geometric sequence, $a_{n}=0.01\cdot 2^{n-1}$
The total amount saved after n days is the partial sum of the geometric sequence,
$ S_{n}=a\displaystyle \cdot\frac{1-r^{n}}{1-r}$
After 30 days,
$ S_{30}=0.01\displaystyle \times\frac{1-2^{30}}{1-2}$
$=0.01(2^{30}-1)=\$ 10,737,418.23$
To reach a billion ($10^{9}$) dollars,
we want n such that $S_{n}=10^{9}$:
$0.01(2^{n}-1)=10^{9}\qquad/\times 100$
$2^{n}-1=10^{11}$
$2^{n}=10^{11}+1\qquad/$ ... apply log to both sides
$n\log 2\approx 11\qquad/\div\log 2$
$ n\approx$36.5412
(the next integer is 37, since 36 is not enough)
It would take 37 days.
