The peak of Genovia's epic rainstorm of $2018$ occurred between $4$ pm$,$ Aug $11,$ and $4$ am$,$ Aug $12.$ During those $12$ hours the rate of rainfall can be modeled by $r(t)=\frac{200e^{\frac{2}{3}}}{(60+e^{\frac{2}{3}}{)}^2}$ in inches per hour, where $t=0$ represenits $4$ pm on Aug $11.$

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

$($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 Aug $11$

$R(4)$ represents the rainfall at $8pm$ on Aug $11$

$R(4)$ represents the rainfall at $4$ pm on Aug $11$

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

$R(x)$ represents the accumulated rainfall $x$ hours from $4$ pm on Aug $11$

$R(x)$ represents the rainfall at $x$ hours after $4$ pm on Aug $11$

$R(x)$ represents the hours after $4$ pm on Aug $11$

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

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

$($d$)$ Which of the following is NOT correct?

$r(t)$ determines the rate of change of the rainfall

$R(x)$ is the same as the area of the region under the graph of $r(t)$ from $0$ to $x$

$R(x)$ is the derivative of $r(t)$

$R(x)$ represents the total change of rainfall $x$ hours after $4pm$ on Aug $11$

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