$($c$)$ Simon now decides to use the engine more realistically to fly a rocket vertically upwards with speed given by

$v=at$ for $t$ the time in seconds, where the acceleration $a$ is assumed to be constant.

Simon wants to find the maximum possible value of $T$ by controlling the acceleration a via the engine design. Finding

the value of $T$ from $(*)$ for a given value of a will involve solving the cubic equation

$Y=\frac{23}{3}{a}^2{T}^3-\frac{9}{2}a{T}^2+1T.$

Simon realises that one can find a maximum value for $T$ by using implicit differentiation. Writing $T$ as $T(a)$ and

differentiating, Simon finds an equation one can solve to give exactly one non$-$zero value of $T($as a function of $a)$

where $d\frac{T}{d}a=0.$ What is this value of $T?$

Answer: $T(a)=$

$($d$)$ From this value of $T(a),$ Simon now solves equation $(**)$ to find the acceleration a that allows the rocket to run

for the maximum time. What is the value of $a($as a function of $Y)$ in this case?