The vertical position, in meters, of a hot-air balloon above ground level is modeled by the differentiable function
h(t)=−0.02t3+0.30t2+5.0,
where t is the time in minutes after lift-off for 0≤t≤12.
(a) Compute h′(t) and interpret its meaning in the context of the balloon flight, including units.
(b) Evaluate h′(5) and h′(8), and determine at which of those two times the balloon is rising more rapidly. Show all work, including units.
(c) Between t=5 and t=8 minutes, the ground crew claims the balloon’s rate of ascent increased by more than 20\%. Verify or refute this claim using your answers from part (b), and justify your reasoning quantitatively.