Answer
$$a = 10.82 \space in$$
Work Step by Step
1)Maximum applied load (Limited by compressive strength of steel)
$\sigma_{st} = \frac{P_{max}}{11.7\space in^2} = 30\space ksi$
Solving for $P_{max}$ :
$P_{max} = 351\space kip$
2)Bearing Plate dimensions (limited by compressive strength of concrete):
$\sigma_{conc} = \frac{P_{max}}{A_{PLATE}} = \frac{351\space kip}{a^2} = 3.0 \space ksi$
Solving for unknown quantity, a:
$a = \sqrt{\frac{351\space kip}{3.0\space ksi}} = \boxed{10.82\space in}\space\space\leftarrow ANS$
