A specialized sensor measures the pressure PPP, in kilopascals (kPa), within a experimental vessel. The pressure variation over time ttt, in seconds, is modeled by the function:
P(t)=2(t+2)(t−1)(t−4) P(t) = 2(t + 2)(t - 1)(t - 4) P(t)=2(t+2)(t−1)(t−4)A technician applies a vertical calibration offset to the readings. Given that the graph of the adjusted pressure y=P(t)−ky = P(t) - ky=P(t)−k passes through the point with coordinates (0,−4)(0, -4)(0,−4), find the value of the constant kkk.
The experiment is repeated with a time-delay mmm. Given that the curve with equation y=P(t+m)y = P(t + m)y=P(t+m) passes through the origin (0,0)(0, 0)(0,0), determine the three possible values of the constant mmm.
Determine an expression for P′(t)P'(t)P′(t), the rate of change of pressure with respect to time.
Hence find the set of values of ttt for which the rate of change of pressure is less than 6 kPa/s6 \text{ kPa/s}6 kPa/s.
