Problem $7$: $(10$ points$)$ The starship Voyager has accidentally stumbled into the sticky $($and deadly$)$ Toffee Nebula. Objects moving through the Toffee Nebula decelerate at a rate proportional to the square of their velocity multiplied by the time they have spent in the nebula:

$\backslash [$

$\backslash$frac$\{$d v$\}\{$d t$\}=-$k t v$^\{2\}$

$\backslash ]$

for some positive constant $\backslash ($ k $\backslash ).$ Suppose that Voyager's initial velocity is $50$ gigameters per second and that $\backslash ($ k$=9\backslash ).$

a$.(4$ points$)$ Solve the initial value problem for $\backslash ($ v$($t$)\backslash ),$ Voyager's velocity function.

b$.(4$ points$)$ Find $\backslash ($ d$($t$)\backslash ),$ the total distance Voyager has traveled through the nebula at time $\backslash ($ t $\backslash ).$ You may assume that $\backslash ($ d$(0)=0\backslash ).$ You may need to use an integration table for this problem. You may also assume that Voyager's velocity is always positive, so distance traveled is the integral of velocity.

c$.(2$ points$)$ If the nebula is $5$ gigameters wide, will Voyager escape the nebula? Hint: Take the limit $\backslash (\backslash$lim $\_\{$t $\backslash$rightarrow $\backslash$infty$\}$ d$($t$)\backslash )$ and determine if that limit is $\backslash (>5\backslash )$ or not.