The peak of Genovia's epic rainstorm of $2014$ occurred between $4pm,$ Sept $13,$ and $4am,$ Sept $14.$ During those $12$ hours the rate of rainfall can be modeled by $r(t)=\frac{200e^{\frac{2t}{3}}}{(60+e^{\frac{2t}{3}}{)}^2}$ in inches per hour, where $t=0$ represents $4pm$ on Sept $13.$

Let $R(x)={\displaystyle \underset{0}{\overset{x}{}}}\frac{200e^{\frac{2t}{3}}}{(60+e^{\frac{2t}{3}}{)}^2}dt$

$($a$)$ Calculate $R(4).$ Round to $2$ decimal places if needed. What does $R(4)$ represent in the context? $R(4)=,$ which represents

$R(4)$ represents the accumulated rainfall from $4pm$ to $8pm$ on Sept $13$

$R(4)$ represents the rainfall at $4pm$ on Sept $13$

$R(4)$ represents the rainfall at $8pm$ on Sept $13$

$($b$)$ What does $R(x)$ represent?

$R(x)$ represents the accumulated rainfall $x$ hours from $4pm$ on Sept $13$

$R(x)$ represents the rainfall at $x$ hours after $4pm$ on Sept $13$

$R(x)$ represents the hours after $4pm$ on Sept $13$

$($c$)$ Where is $R(x)$ changing the fastest? Round to the whole number.

$R(x)$ is changing the fastest at $t=$