Answer
$\Delta t_p=9.842s$
Work Step by Step
Proper time and dilated time are related using the formula $$\Delta t'=\gamma \Delta t_p$$ Therefore, the proper time must be equal to $$\Delta t_p=\frac{\Delta t'}{\gamma}$$ To find gamma, use the formula $$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$ where $v=\beta c$. Substitute the known value of $\beta=0.7705$ into the equation to get $$\gamma=\frac{1}{\sqrt{1-0.7705^2}}=1.569$$ Substituting the known values of $\Delta t'=15.44s$ and $\gamma=1.569$ yields a proper time of $$\Delta t_p=\frac{15.44s}{1.569}=9.842s$$
