Answer
The time interval is $~~2.58\times 10^{-5}~s$
Work Step by Step
We can find $\gamma$:
$\gamma = \frac{1}{\sqrt{1-\beta^2}}$
$\gamma = \frac{1}{\sqrt{1-0.250^2}}$
$\gamma = 1.033$
We can find the temporal coordinate $t'$ of the big flash:
$t' = \gamma~(t-vx/c^2)$
$t' = 1.033~[(0)-\frac{(0.250)(3.0\times 10^8~m/s)(0)}{(3.0\times 10^8~m/s)^2}]$
$t' = 0$
We can find the temporal coordinate $t'$ of the small flash:
$t' = \gamma~(t-vx/c^2)$
$t' = 1.033~[(0)-\frac{(0.250)(3.0\times 10^8~m/s)(3.00\times 10^4~m)}{(3.0\times 10^8~m/s)^2}]$
$t' = (1.033)(-2.50\times 10^{-5}~s)$
$t' = -2.58\times 10^{-5}~s$
The time interval is $~~2.58\times 10^{-5}~s$
