$4.$ The displacement thickness $\backslash (\backslash$delta$^\{*\}\backslash )$ represents the distance that streamlines are displaced due to the development of a boundary layer. For a given duct, this thickness acts as a downstream area contraction of the core flow. Consider a wind tunnel test section that has a constant $\backslash (45\backslash$mathrm$\{$~cm$\}\backslash$times $45\backslash$mathrm$\{$~cm$\}\backslash )$ cross$-$section and is $5$ m long. If the freestream velocity is $\backslash (7\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ at the start of the test section $(\backslash ($ x$=0\backslash )),$ estimate: a$)$ the magnitude of the core velocity gradient $(\backslash (\backslash$mathrm$\{$d$\}$ u $/\backslash$mathrm$\{$d$\}$ x $\backslash ))$ between the start and end of the test section. $(3$ points$)$ b$)$ If $\backslash (\backslash$mathrm$\{$d$\}$ u $/\backslash$mathrm$\{$d$\}$ x$=0\backslash )$ is desired along the entire test section length, approximate the angle that the test section walls should be set at$.(2$ points$)$