A temperature difference between two objects is modeled (in degrees Celsius) by T(t)=t3−6t2+2 for 0≤t≤4, where t is measured in hours. The time t at which T(t)=0 represents thermal equilibrium.
(a) Show that T(0) and T(4) have opposite signs and explain what this implies about the existence of an equilibrium time in the interval (0,4).
(b) Determine the minimum number of equal halving steps, beginning with [0,4], required to guarantee that the length of the bracketing interval for the equilibrium time is at most 0.1 hour. Clearly state your reasoning.
(c) Carry out the number of halvings found in part (b), each time choosing the half-interval that still contains the equilibrium time. State the resulting interval and give an approximation of the equilibrium time to the nearest tenth of an hour.
(d) Explain why your approximation in part (c) is guaranteed to be within 0.05 hour of the exact equilibrium time.