Answer
$\beta =0.9959 $
Work Step by Step
First we'll use the time dilation equation and solve for Beta;
$\Delta t=\Delta t_0 / \sqrt{1-\beta^2 }$
$\sqrt{1-\beta^2 } \Delta t=\Delta t_0 $
$\sqrt{1-\beta^2 } =\Delta t_0 / \Delta t $
$1-\beta^2 =(\Delta t_0 / \Delta t)^2 $
$\beta =\sqrt{1-(\Delta t_0 / \Delta t)^2} $
We were given that the travel time is 4 years. His daughter back on earth must age 40 years and also wait 4 years for her father to return, for a total of 44 years from her perspective. Therefore;
$\beta =\sqrt{1-(4years / 44years)^2} $
$\beta =0.9959 $
