A high-precision thermal sensor monitors the interaction between a heating probe and its surrounding coolant jacket. The probe's temperature H(t)H(t)H(t), in Celsius, is modeled by H(t)=6et−3+8H(t) = 6e^{t-3} + 8H(t)=6et−3+8, and the jacket's internal boundary temperature C(t)C(t)C(t) is modeled by C(t)=28−t2C(t) = 28 - t^2C(t)=28−t2, where t t\,t is the time in minutes (t∈Rt \in \mathbb{R}t∈R).
Point P P\,P represents a state on the heating curve H(t)H(t)H(t) where the temperature is exactly 32∘C32^\circ\text{C}32∘C.
Find the value of t t\,t at point PPP, writing your answer in the form lnk+3\ln k + 3lnk+3, where k k\,k is a constant to be found.
The temperatures of the probe and the jacket are equal at times t=αt = \alphat=α and t=βt = \betat=β.
Using a suitable interval and a suitable function that should be stated, show that α=3.367\alpha = 3.367α=3.367 correct to 3 decimal places.
The iterative equation
tn+1=−20−6etn−3 t_{n+1} = -\sqrt{20 - 6e^{t_n-3}} tn+1=−20−6etn−3is used to find an approximation for the second intersection time, β\betaβ.
Using this formula with t1=−4.4t_1 = -4.4t1=−4.4,
find the value of t2 t_2\,t2 and the value of β\betaβ, giving each answer to 6 decimal places.
