HarMathAp$1210\mathrm{.}4\mathrm{.}00$d

Analysis of daily output of a factory during a $8-$hour shift shows that the hourly number of units $y$ produced after $t$ hours of production is

$y=102t\frac{1}{2}{t}^2-t^3,0t8$

$($a$)$ After how many hours will the hourly number of units be maximized?

$$ hr

$($b$)$ What is the maximum hourly output?

$$ units$/$hr