A $13\mathrm{.}0$ kg monkey climbs up a $116$ N uniform ladder of length $\backslash ($ L $\backslash ),$ as shown in Figure P$12\mathrm{.}41.$ The upper and lower ends of the ladder rest on frictionless surfaces. The lower end is fastened to the wall by a horizontal rope.

$($a$)$ Find the tension in the rope when the monkey is one third of the way up the ladder.

N

$($b$)$ If the rope can support a maximum tension of $99$ N $,$ find the maximum distance $\backslash ($ d $\backslash )$ that the monkey can climb up $($along$)$ the ladder before the rope breaks. Express your answer as a function of $\backslash ($ L $\backslash )($e$.$g$.,0\mathrm{.}33*$L$).$