Suppose the position of an object $($as a function of time$)$ is given by $r(t)=($:$t^2,9-5t,2t$:$).$

$[2]($a$)$ At what time does this curve cross through the $xz$ plane?

$[6]($b$)$ Find simplified expressions for the tangential acceleration $a_T,$ the normal acceleration $a_N,$ and the curvature $$ at an arbitrary time $t.$

$[4]($c$)$ By hand, find the distance this object travels from $t=0$ to $t=3.$ Give a simplified answer involving ln$.$ You may use this formula:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\sqrt[\phantom{2}]{a^2+x^2}dx=\frac{x}{2}\sqrt[\phantom{2}]{a^2+x^2}+\frac{a^2}{2}ln(x+\sqrt[\phantom{2}]{a^2+x^2})+C$