I thought this was going to involve logarithms, lol.
It suffices to maximize . The rectangle must have a circumscribing circle of radius no more than 14 inches in order to fit in the log. The radius of the circumscribing circle is half the diagonal length. This comes to the condition . If this does not have equality, we could increase either the width or the height until it does, thus increasing the strength of the beam. Therefore, we must have or . Substituting, you must maximize . Can you take it now?