A signal processor monitors the power P P\,P in milliwatts of an oscillating beam over time t t\,t milliseconds. The power output is modeled by the function
P(t)=30−5∣4−t∣,t≥0P(t) = 30 - 5|4 - t|, \quad t \ge 0P(t)=30−5∣4−t∣,t≥0
Determine the value of PP(2)PP(2)PP(2).
Solve the equation P(t)=5tP(t) = 5tP(t)=5t.
Given that the equation P(t)=kP(t) = kP(t)=k has exactly two roots, state the range of possible values for the constant kkk.
The power signal is adjusted such that the new power output Q(t)Q(t)Q(t) is given by Q(t)=aP(t−b)Q(t) = aP(t - b)Q(t)=aP(t−b). The maximum power of the new signal is 6 mW and occurs at t=12t = 12t=12. Find the value of the constants a a\,a and bbb.
