Problem $1(25$ points$)$:

A hot$-$air balloon is held with three ropes $AB,AC$ and AD as shown in Fig$\mathrm{.}1.$ The rope will break when it is subjected to a tensile force of $450$ N $.$

$($a$)$ Express the tensile forces, $T_{AB},T_{AC}$ and $T_{AD}$ in the ropes in a Cartesian vector form. $(5\%)$

$($b$)$ Write the equilibrium equation of the system when the hot$-$air balloon is subjected to an uplift force F$.(5\%)$

$($c$)$ Determine the maximum uplift force $F$ the balloon can resist if rope AD is assumed to break first. $(10\%)$

$($d$)$ Verify if the assumption in $($c$)$ is correct or not. $(5\%)$