Answer
$t = 2.01~\frac{D}{c}$
Work Step by Step
We can find $\beta$:
$\gamma = \frac{1}{\sqrt{1-\beta^2}}$
$\sqrt{1-\beta^2} = \frac{1}{\gamma}$
$1-\beta^2 = \frac{1}{\gamma^2}$
$\beta^2 = 1-\frac{1}{\gamma^2}$
$\beta = \sqrt{1-\frac{1}{\gamma^2}}$
$\beta = \sqrt{1-\frac{1}{10^2}}$
$\beta = 0.994987$
We can find the travel time as measured from the rest frame of the Earth:
$t = \frac{2D}{0.994987~c}$
$t = 2.01~\frac{D}{c}$
