This question builds on an earlier problem. The randomized numbers may have changed, but have your work for the previous problem available to help with this one. in the third picture.

$($a$)$ Express the $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ coordinates of point $\backslash ($ A $\backslash )$ as functions of $\backslash ($ t $\backslash )$ :

$\backslash ($ x$=\backslash )$ and $\backslash ($ y$=\backslash )$

$($b$)$ Write a formula for the slope of the tangent line to the circle at the point $\backslash ($ A $\backslash )$ at time $\backslash ($ t $\backslash )$ seconds:

$($c$)$ Express the $\backslash ($ x $\backslash )-$coordinate of the right end of the rod at point $\backslash ($ B $\backslash )$ as a function of $\backslash ($ t $\backslash )$ :

$($d$)$ Express the slope of the rod as a function of $\backslash ($ t: $\backslash )\backslash$begin$\{$tabular$\}\{|\}$

$\backslash (-\backslash$frac$\{1\}\{2\}\backslash )\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$($c$)$ Find the first time when the rod is tangent to the circle:

$($d$)$ At the time in $($c$),$ what is the slope of the rod?

$($e$)$ Find the second time when the rod is tangent to the circle: $\backslash (\backslash$frac$\{1\mathrm{.}107\}\{5\backslash$pi$\}\backslash )$ seconds. $($f$)$ At the time in $($e$),$ what is the slope of the rod?