Problem $1$: The beam shown below has been nailed together from $7$ identical pieces of wood with dimensions $\backslash (\backslash$mathrm$\{$t$\}=0\mathrm{.}75\backslash )$ in and $\backslash ($ w$=4\backslash$mathrm$\{$in$\}\backslash ).$ A point load, $\backslash ($ P $\backslash ),$ is applied at $\backslash ($ x$=5\backslash$mathrm$\{$ft$\}\backslash ),$ and the distributed load shown below acts over distance of $1$ ft $.$ You may neglect the weight of the beam. Assuming the shear force a single nail can transfer without failing is $75$ lb :

$($a$)$ What is the maximum allowable spacing, $\backslash (\backslash$mathrm$\{$s$\}\_\{12\}\backslash ),$ between the nails used to hold pieces $1$ and $2$ together?

$($b$)$ What is the maximum allowable spacing, $\backslash (\backslash$mathrm$\{$s$\}\_\{23\}\backslash ),$ between the nails used to hold pieces $2$ and $3$ together?