$5\mathrm{.}6-16$ A small balcony constructed of wood is supported by three identical cantilever beams $($see figure$).$ Each beam has length $L_1=2\mathrm{.}1m,$ width $b,$ and height $h=4\frac{b}{3}.$ The dimensions of the balcony floor are $L_1{L}_2,$ where $L_2=2\mathrm{.}5m.$ The design load is $5\mathrm{.}5$ kPa acting over the entire floor area. $($This load accounts for all loads except the weights of the cantilever beams, which have a weight density $=5\mathrm{.}5k\frac{N}{m^3}.)$ The allowable bending stress in the cantilevers is $15$ MPa $.$

Assuming that the middle cantilever supports $50\%$ of the load and each outer cantilever supports $25\%$ of the load, determine the required dimensions $b$ and $h.$