In 10 days, the earth picks up

Let the amount of dust picked up by the earth in 45 days be x pounds.

Since the amount of dust picked up by the earth and the number of days are in direct variation, we have:

Ratio of the dust picked up by the earth in pounds = ratio of the number of days taken

$\Rightarrow \frac{10}{45}=\frac{2.6 \times 10^{8}}{x}$

$\Rightarrow x \times 10=45 \times 2.6 \times 10^{8}$

$\Rightarrow x=\frac{45 \times 2.6 \times 10^{8}}{10}$

$=\frac{117 \times 10^{8}}{10}$

$=11.7 \times 10^{8}$

Thus, $11.7 \times 10^{8}$ pounds of dust will be picked up by the earth in 45 days.