A cup of coffee cools according to the law ( dT dt =-k(T-20) )
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A cup of coffee cools according to the law dtdT=−k(T−20), where T(t) is the temperature (in °C) at time t minutes, and k>0 is a constant. Initially the coffee has temperature T(0)=92°C, and k=0.04.
(a) Starting from t=0, use a uniform time step of Δt=5 minutes to obtain a numerical estimate of the coffee temperature at t=10 minutes. Show the sequence of ordered pairs (t,T) generated by your calculation.
(b) Solve the differential equation explicitly to find an exact expression for T(t). Then determine the exact temperature of the coffee at t=10 minutes.
(c) Compute the absolute error and the percentage error of the estimate from part (a).
(d) Suppose you wish the absolute error at t=10 minutes to be no greater than 0.2 °C. Without performing additional numerical calculations, explain whether decreasing the time step to Δt=1 minute is sufficient to guarantee this level of accuracy, citing evidence from the behavior of the differential equation.