$(9$ points$)$ Nevada State operates a septic tank sitting atop the Dawson building. The rate of sewage released from the tank at time $t$ is modeled by the function

$R(t)=-2+5t$

Suppose the supply of fresh sewage pumped into the septic tank is modeled by the function

$S(t)=\frac{15t}{1+3t}$

At $t=0,$ the Dawson tank holds $1000$ gallons of sewage.

$($a$)$ Write the definite integral measuring the total amount of sewage removed from the tank over the period $0t6.$ You do not need to evaluate the integral.

$($b$)$ Write the expression for $Y(x),$ the total sewage level in the tank at time $x$ using the second Fundamental Theorem of Calculus.

$($c$)$ What is the instantaneous rate of change of sewage in the septic tank at $x=4?$

$($d$)$ When does the amount of sewage in the septic tank hit its minimum? Fully justify your answer.

$($e$)$ What is the long$-$term projection of the amount of sewage in the septic tank? Is the septic tank a sustainable solution for the campus of Nevada State?

$\frac{1}{2}$

Solve for all parts a through e