1.1. The altitude of a hot air balloon, $a$ feet, $t$ minutes after it begins its ascent is modelled by the equation

$a=8{t}^{3}-132{t}^{2}+726t0\leqt\leq11$

According to the model,

(i) find the cruising altitude of the balloon, given that it takes exactly 11 minutes to reach this altitude,

(ii) show that the balloon rises by exactly 250 feet between $t=3$ and $t=8$, and hence suggest how the pilot flew the balloon during its ascent.

1.2. Show that there is only one real solution to the equation $a=0$, and hence explain why the model cannot be used indefinitely.