The temperature of an industrial cooling unit, TCT_CTC, is modelled by the equation TC=3et−4+12T_C = 3e^{t-4} + 12TC=3et−4+12, where t t\,t is the time in hours after a process starts. The surrounding ambient temperature, TAT_ATA, is modelled by TA=40−t2T_A = 40 - t^2TA=40−t2.
Point P P\,P represents a specific state where the unit temperature reaches 42∘C42^\circ\text{C}42∘C.
Determine the time t t\,t at which point P P\,P occurs, giving your answer in the form lnk+4\ln k + 4lnk+4, where k k\,k is an integer to be found.
The cooling unit temperature TC T_C\,TC and ambient temperature TA T_A\,TA are equal at times t=αt = \alphat=α and t=βt = \betat=β, where α>β\alpha > \betaα>β.
By choosing a suitable interval and a function f(t)f(t)f(t) which should be stated, show that α=4.691\alpha = 4.691α=4.691 correct to 3 decimal places.
The iterative formula
tn+1=−28−3etn−4 t_{n+1} = -\sqrt{28 - 3e^{t_n-4}} tn+1=−28−3etn−4is used to approximate the earlier time β\betaβ.
Using this formula with t1=−5t_1 = -5t1=−5, find the value of t2 t_2\,t2 and the value of β\betaβ, giving each answer to 6 decimal places.