A laboratory tracks the pressure variance VVV, in kilopascals, within a reaction vessel. The variance is modelled by the function V(t)=(t+2)(t−4)(2t−5)V(t) = (t+2)(t-4)(2t-5)V(t)=(t+2)(t−4)(2t−5), where t t\,t is the time in minutes relative to a baseline event.
A secondary model is defined as g(t)=12V(t−3)+k\displaystyle g(t) = \frac{1}{2}V(t - 3) + kg(t)=21V(t−3)+k. Given that the graph of y=g(t)y = g(t)y=g(t) passes through the point (3,14)(3, 14)(3,14), determine the value of the constant kkk.
The pressure variance model is shifted horizontally such that h(t)=V(t+m)h(t) = V(t + m)h(t)=V(t+m). Given that the graph of y=h(t)y = h(t)y=h(t) passes through the origin (0,0)(0,0)(0,0), find all possible values of the constant mmm.
Find V′(t)V'(t)V′(t).
Determine the set of values of t t\,t for which the rate of change of the pressure variance is less than -6.
