Answer
$\Delta x' = \gamma~[(400~m)-c\beta~(1.00\times 10^{-6}~s)]$
Work Step by Step
We can write an expression for $x_A'$:
$x_A' = \gamma~(x_A-c\beta~t_A)$
We can write an expression for $x_B'$:
$x_B' = \gamma~(x_B-c\beta~t_B)$
We can write an expression for $\Delta x'$:
$\Delta x' = x_B'-x_A'$
$\Delta x' = \gamma~[(x_B-x_A)-c\beta~(t_B-t_A)]$
$\Delta x' = \gamma~[(400~m)-c\beta~(1.00\times 10^{-6}~s)]$
